Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.9% → 78.9%

Time: 28.1s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% 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: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-113}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D_m}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, {\left(\left(\left(M \cdot D_m\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D_m}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -2.2e-113)
   (*
    (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D_m d) (* M 0.5)) 2.0)) l))))
   (if (<= d -1.06e-306)
     (*
      (* (- d) (sqrt (/ (/ 1.0 h) l)))
      (fma -0.5 (pow (* (* (* M D_m) (/ 0.5 d)) (sqrt (/ h l))) 2.0) 1.0))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* (/ D_m d) 0.5)) 2.0))))
      (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -2.2e-113) {
		tmp = (sqrt((d / h)) * (sqrt(-d) / sqrt(-l))) * (1.0 - (0.5 * ((h * pow(((D_m / d) * (M * 0.5)), 2.0)) / l)));
	} else if (d <= -1.06e-306) {
		tmp = (-d * sqrt(((1.0 / h) / l))) * fma(-0.5, pow((((M * D_m) * (0.5 / d)) * sqrt((h / l))), 2.0), 1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -2.2e-113)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D_m / d) * Float64(M * 0.5)) ^ 2.0)) / l))));
	elseif (d <= -1.06e-306)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l))) * fma(-0.5, (Float64(Float64(Float64(M * D_m) * Float64(0.5 / d)) * sqrt(Float64(h / l))) ^ 2.0), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D_m / d) * 0.5)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -2.2e-113], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.06e-306], N[(N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.20000000000000004e-113

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

      1. associate-*r/ 77.4%

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

   5. Applied egg-rr 77.4%

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

      1. frac-2neg 77.4%

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

      2. sqrt-div 83.5%

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

   7. Applied egg-rr 83.5%

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

   if -2.20000000000000004e-113 < d < -1.06e-306

   1. Initial program 43.3%

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

   3. Simplified 43.2%

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

      1. associate-*r/ 43.4%

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

   5. Applied egg-rr 43.4%

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

      1. expm1-log1p-u 20.7%

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

      2. expm1-udef 11.0%

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

   7. Applied egg-rr 10.8%

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

      1. expm1-def 16.1%

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

      2. expm1-log1p 31.4%

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

      3. +-commutative 31.4%

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

      4. fma-def 31.4%

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

      5. associate-*r/ 31.4%

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

      6. associate-*l/ 31.3%

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

      7. *-commutative 31.3%

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

      8. associate-*r/ 31.3%

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

      9. *-commutative 31.3%

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

      10. associate-/l* 31.4%

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

      11. associate-*r/ 31.3%

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

   9. Simplified 31.3%

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

       1. add-sqr-sqrt 31.3%

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

       2. pow2 31.3%

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

       3. sqrt-prod 31.3%

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

       4. unpow2 31.3%

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

       5. sqrt-prod 21.7%

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

       6. add-sqr-sqrt 34.8%

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

       7. associate-*r* 34.9%

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

   11. Applied egg-rr 34.9%

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

   12. Taylor expanded in d around -inf 67.1%

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

       1. mul-1-neg 67.1%

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

       2. *-commutative 67.1%

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

       3. distribute-rgt-neg-in 67.1%

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

       4. associate-/r* 67.1%

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

   14. Simplified 67.1%

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

   if -1.06e-306 < d

   1. Initial program 59.8%

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

   3. Simplified 59.2%

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

   5. Applied egg-rr 31.1%

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

      1. expm1-def 46.7%

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

      2. expm1-log1p 77.9%

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

      3. associate-*l* 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*l* 77.9%

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

   7. Simplified 77.9%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{-113}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, {\left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 2: 77.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 + ((h / l) * ((((m / 2.0d0) * (d_m / d)) ** 2.0d0) * (-0.5d0)))))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * ((m * ((d_m / d) * 0.5d0)) ** 2.0d0)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

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

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if h <= -5e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 + ((h / l) * (math.pow(((M / 2.0) * (D_m / d)), 2.0) * -0.5))))
	else:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (math.sqrt(h) * math.sqrt(l)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 59.7%

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

   3. Simplified 59.7%

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

      1. frac-2neg 36.1%

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

      2. sqrt-div 44.0%

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

   5. Applied egg-rr 73.9%

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

   if -4.999999999999985e-310 < h

   1. Initial program 60.3%

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

   3. Simplified 59.6%

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

   5. Applied egg-rr 31.4%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 78.5%

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

      3. associate-*l* 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-*l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 76.1%

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

Alternative 3: 77.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-5d-310)) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_m / 2.0d0) * (m / d)) ** 2.0d0)))) * ((sqrt(-d) / sqrt(-h)) * sqrt((d / l)))
    else
        tmp = (1.0d0 - (0.5d0 * ((h / l) * ((m * ((d_m / d) * 0.5d0)) ** 2.0d0)))) * (d / (sqrt(h) * sqrt(l)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if h < -4.999999999999985e-310

   1. Initial program 59.7%

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

   3. Simplified 59.7%

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

      1. frac-2neg 36.1%

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

      2. sqrt-div 44.0%

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

   5. Applied egg-rr 72.8%

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

   if -4.999999999999985e-310 < h

   1. Initial program 60.3%

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

   3. Simplified 59.6%

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

   5. Applied egg-rr 31.4%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 78.5%

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

      3. associate-*l* 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-*l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 75.6%

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

Alternative 4: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.9 \cdot 10^{+195}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D_m}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, {\left(\left(\left(M \cdot D_m\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D_m}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -2.9e+195)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D_m d) (* M 0.5)) 2.0)) l)))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (if (<= h -5e-310)
     (*
      (* (- d) (sqrt (/ (/ 1.0 h) l)))
      (fma -0.5 (pow (* (* (* M D_m) (/ 0.5 d)) (sqrt (/ h l))) 2.0) 1.0))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* (/ D_m d) 0.5)) 2.0))))
      (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -2.9e+195) {
		tmp = (1.0 - (0.5 * ((h * pow(((D_m / d) * (M * 0.5)), 2.0)) / l))) * (sqrt((d / l)) * sqrt((d / h)));
	} else if (h <= -5e-310) {
		tmp = (-d * sqrt(((1.0 / h) / l))) * fma(-0.5, pow((((M * D_m) * (0.5 / d)) * sqrt((h / l))), 2.0), 1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -2.9e+195)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D_m / d) * Float64(M * 0.5)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l))) * fma(-0.5, (Float64(Float64(Float64(M * D_m) * Float64(0.5 / d)) * sqrt(Float64(h / l))) ^ 2.0), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D_m / d) * 0.5)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -2.9e+195], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(M * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.89999999999999992e195

   1. Initial program 50.3%

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

   3. Simplified 50.2%

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

      1. associate-*r/ 59.0%

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

   5. Applied egg-rr 59.0%

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

   if -2.89999999999999992e195 < h < -4.999999999999985e-310

   1. Initial program 63.0%

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

   3. Simplified 63.1%

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

      1. associate-*r/ 64.1%

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

   5. Applied egg-rr 64.1%

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

      1. expm1-log1p-u 36.7%

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

      2. expm1-udef 25.4%

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

   7. Applied egg-rr 20.0%

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

      1. expm1-def 27.9%

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

      2. expm1-log1p 49.6%

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

      3. +-commutative 49.6%

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

      4. fma-def 49.6%

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

      5. associate-*r/ 49.6%

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

      6. associate-*l/ 49.6%

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

      7. *-commutative 49.6%

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

      8. associate-*r/ 49.6%

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

      9. *-commutative 49.6%

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

      10. associate-/l* 50.7%

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

      11. associate-*r/ 49.6%

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

   9. Simplified 49.6%

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

       1. add-sqr-sqrt 49.6%

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

       2. pow2 49.6%

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

       3. sqrt-prod 49.6%

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

       4. unpow2 49.6%

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

       5. sqrt-prod 32.4%

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

       6. add-sqr-sqrt 52.6%

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

       7. associate-*r* 53.5%

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

   11. Applied egg-rr 53.5%

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

   12. Taylor expanded in d around -inf 77.9%

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

       1. mul-1-neg 77.9%

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

       2. *-commutative 77.9%

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

       3. distribute-rgt-neg-in 77.9%

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

       4. associate-/r* 77.9%

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

   14. Simplified 77.9%

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

   if -4.999999999999985e-310 < h

   1. Initial program 60.3%

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

   3. Simplified 59.6%

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

   5. Applied egg-rr 31.4%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 78.5%

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

      3. associate-*l* 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-*l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.9 \cdot 10^{+195}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, {\left(\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 5: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -5.8 \cdot 10^{+224}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D_m \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D_m}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -5.8e+224)
   (*
    (sqrt (/ (* d (/ d h)) l))
    (+ 1.0 (* -0.5 (* h (/ (pow (/ (* D_m 0.5) (/ d M)) 2.0) l)))))
   (if (<= h -5e-310)
     (*
      (* (- d) (sqrt (/ (/ 1.0 h) l)))
      (fma -0.5 (* (/ h l) (pow (* D_m (* M (/ 0.5 d))) 2.0)) 1.0))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* (/ D_m d) 0.5)) 2.0))))
      (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -5.8e+224) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (h <= -5e-310) {
		tmp = (-d * sqrt(((1.0 / h) / l))) * fma(-0.5, ((h / l) * pow((D_m * (M * (0.5 / d))), 2.0)), 1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -5.8e+224)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(D_m * 0.5) / Float64(d / M)) ^ 2.0) / l)))));
	elseif (h <= -5e-310)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l))) * fma(-0.5, Float64(Float64(h / l) * (Float64(D_m * Float64(M * Float64(0.5 / d))) ^ 2.0)), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D_m / d) * 0.5)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -5.8e+224], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -5.79999999999999978e224

   1. Initial program 50.5%

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

   3. Simplified 50.4%

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

      1. associate-*r/ 63.7%

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

   5. Applied egg-rr 63.8%

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

      1. expm1-log1p-u 35.9%

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

      2. expm1-udef 18.5%

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

   7. Applied egg-rr 9.6%

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

   9. Simplified 59.3%

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

   if -5.79999999999999978e224 < h < -4.999999999999985e-310

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

      1. associate-*r/ 62.6%

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

   5. Applied egg-rr 62.6%

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

      1. expm1-log1p-u 35.4%

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

      2. expm1-udef 23.6%

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

   7. Applied egg-rr 18.8%

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

      1. expm1-def 27.6%

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

      2. expm1-log1p 48.7%

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

      3. +-commutative 48.7%

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

      4. fma-def 48.7%

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

      5. associate-*r/ 48.7%

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

      6. associate-*l/ 48.7%

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

      7. *-commutative 48.7%

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

      8. associate-*r/ 48.7%

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

      9. *-commutative 48.7%

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

      10. associate-/l* 49.7%

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

      11. associate-*r/ 48.7%

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

   9. Simplified 48.7%

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

   10. Taylor expanded in d around -inf 71.5%

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

       1. mul-1-neg 74.2%

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

       2. *-commutative 74.2%

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

       3. distribute-rgt-neg-in 74.2%

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

       4. associate-/r* 74.2%

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

   12. Simplified 71.5%

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

   if -4.999999999999985e-310 < h

   1. Initial program 60.3%

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

   3. Simplified 59.6%

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

   5. Applied egg-rr 31.4%

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

      1. expm1-def 47.1%

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

      2. expm1-log1p 78.5%

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

      3. associate-*l* 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-*l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5.8 \cdot 10^{+224}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 6: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(D_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{-110}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot t_0}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot t_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D_m}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (pow (* D_m (* M (/ 0.5 d))) 2.0)))
   (if (<= d -1.4e-110)
     (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* 0.5 (/ (* h t_0) l))))
     (if (<= d -1.06e-306)
       (* (* (- d) (sqrt (/ (/ 1.0 h) l))) (fma -0.5 (* (/ h l) t_0) 1.0))
       (*
        (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* (/ D_m d) 0.5)) 2.0))))
        (/ d (* (sqrt h) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow((D_m * (M * (0.5 / d))), 2.0);
	double tmp;
	if (d <= -1.4e-110) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (0.5 * ((h * t_0) / l)));
	} else if (d <= -1.06e-306) {
		tmp = (-d * sqrt(((1.0 / h) / l))) * fma(-0.5, ((h / l) * t_0), 1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(D_m * Float64(M * Float64(0.5 / d))) ^ 2.0
	tmp = 0.0
	if (d <= -1.4e-110)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))));
	elseif (d <= -1.06e-306)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l))) * fma(-0.5, Float64(Float64(h / l) * t_0), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D_m / d) * 0.5)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Power[N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -1.4e-110], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.06e-306], N[(N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.4e-110

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

      1. associate-*r/ 77.4%

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

   5. Applied egg-rr 77.4%

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

   6. Taylor expanded in M around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. *-commutative 77.4%

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

      3. associate-*r/ 77.4%

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

      4. associate-*l/ 77.4%

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

      5. associate-*r/ 77.4%

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

      6. associate-*r* 77.4%

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

   8. Simplified 77.4%

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

   if -1.4e-110 < d < -1.06e-306

   1. Initial program 43.3%

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

   3. Simplified 43.2%

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

      1. associate-*r/ 43.4%

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

   5. Applied egg-rr 43.4%

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

      1. expm1-log1p-u 20.7%

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

      2. expm1-udef 11.0%

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

   7. Applied egg-rr 10.8%

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

      1. expm1-def 16.1%

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

      2. expm1-log1p 31.4%

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

      3. +-commutative 31.4%

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

      4. fma-def 31.4%

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

      5. associate-*r/ 31.4%

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

      6. associate-*l/ 31.3%

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

      7. *-commutative 31.3%

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

      8. associate-*r/ 31.3%

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

      9. *-commutative 31.3%

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

      10. associate-/l* 31.4%

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

      11. associate-*r/ 31.3%

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

   9. Simplified 31.3%

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

   10. Taylor expanded in d around -inf 65.2%

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

       1. mul-1-neg 67.1%

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

       2. *-commutative 67.1%

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

       3. distribute-rgt-neg-in 67.1%

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

       4. associate-/r* 67.1%

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

   12. Simplified 65.2%

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

   if -1.06e-306 < d

   1. Initial program 59.8%

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

   3. Simplified 59.2%

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

   5. Applied egg-rr 31.1%

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

      1. expm1-def 46.7%

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

      2. expm1-log1p 77.9%

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

      3. associate-*l* 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*l* 77.9%

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

   7. Simplified 77.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-110}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-111}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D_m}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D_m \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D_m}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -2.8e-111)
   (*
    (- 1.0 (* 0.5 (/ (* h (pow (* (/ D_m d) (* M 0.5)) 2.0)) l)))
    (* (sqrt (/ d l)) (sqrt (/ d h))))
   (if (<= d -1.06e-306)
     (*
      (* (- d) (sqrt (/ (/ 1.0 h) l)))
      (fma -0.5 (* (/ h l) (pow (* D_m (* M (/ 0.5 d))) 2.0)) 1.0))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* (/ D_m d) 0.5)) 2.0))))
      (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -2.8e-111) {
		tmp = (1.0 - (0.5 * ((h * pow(((D_m / d) * (M * 0.5)), 2.0)) / l))) * (sqrt((d / l)) * sqrt((d / h)));
	} else if (d <= -1.06e-306) {
		tmp = (-d * sqrt(((1.0 / h) / l))) * fma(-0.5, ((h / l) * pow((D_m * (M * (0.5 / d))), 2.0)), 1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -2.8e-111)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D_m / d) * Float64(M * 0.5)) ^ 2.0)) / l))) * Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))));
	elseif (d <= -1.06e-306)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / h) / l))) * fma(-0.5, Float64(Float64(h / l) * (Float64(D_m * Float64(M * Float64(0.5 / d))) ^ 2.0)), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D_m / d) * 0.5)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -2.8e-111], N[(N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D$95$m / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.06e-306], N[(N[((-d) * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D$95$m * N[(M * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.79999999999999995e-111

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

      1. associate-*r/ 77.4%

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

   5. Applied egg-rr 77.4%

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

   if -2.79999999999999995e-111 < d < -1.06e-306

   1. Initial program 43.3%

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

   3. Simplified 43.2%

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

      1. associate-*r/ 43.4%

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

   5. Applied egg-rr 43.4%

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

      1. expm1-log1p-u 20.7%

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

      2. expm1-udef 11.0%

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

   7. Applied egg-rr 10.8%

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

      1. expm1-def 16.1%

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

      2. expm1-log1p 31.4%

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

      3. +-commutative 31.4%

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

      4. fma-def 31.4%

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

      5. associate-*r/ 31.4%

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

      6. associate-*l/ 31.3%

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

      7. *-commutative 31.3%

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

      8. associate-*r/ 31.3%

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

      9. *-commutative 31.3%

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

      10. associate-/l* 31.4%

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

      11. associate-*r/ 31.3%

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

   9. Simplified 31.3%

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

   10. Taylor expanded in d around -inf 65.2%

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

       1. mul-1-neg 67.1%

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

       2. *-commutative 67.1%

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

       3. distribute-rgt-neg-in 67.1%

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

       4. associate-/r* 67.1%

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

   12. Simplified 65.2%

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

   if -1.06e-306 < d

   1. Initial program 59.8%

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

   3. Simplified 59.2%

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

   5. Applied egg-rr 31.1%

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

      1. expm1-def 46.7%

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

      2. expm1-log1p 77.9%

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

      3. associate-*l* 77.9%

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

      4. *-commutative 77.9%

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

      5. associate-*l* 77.9%

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

   7. Simplified 77.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-111}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{elif}\;d \leq -1.06 \cdot 10^{-306}:\\ \;\;\;\;\left(\left(-d\right) \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5, \frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 8: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{D_m}}{0.5}}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -1.16e+14)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (if (<= l 6.5e-200)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (fma -0.5 (* h (/ (pow (/ M (/ (/ d D_m) 0.5)) 2.0) l)) 1.0))
     (if (<= l 6.2e+132)
       (*
        (/ d (sqrt (* h l)))
        (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D_m d)) 2.0)))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -1.16e+14) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else if (l <= 6.5e-200) {
		tmp = sqrt(((d / l) * (d / h))) * fma(-0.5, (h * (pow((M / ((d / D_m) / 0.5)), 2.0) / l)), 1.0);
	} else if (l <= 6.2e+132) {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.125 * ((h / l) * pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -1.16e+14)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (l <= 6.5e-200)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(-0.5, Float64(h * Float64((Float64(M / Float64(Float64(d / D_m) / 0.5)) ^ 2.0) / l)), 1.0));
	elseif (l <= 6.2e+132)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -1.16e+14], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-200], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 * N[(h * N[(N[Power[N[(M / N[(N[(d / D$95$m), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+132], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.16e14

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in h around 0 39.7%

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

      1. frac-2neg 39.7%

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

      2. sqrt-div 53.3%

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

   6. Applied egg-rr 53.3%

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

   if -1.16e14 < l < 6.5000000000000002e-200

   1. Initial program 68.4%

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

   3. Simplified 66.3%

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

      1. associate-*r/ 71.9%

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

   5. Applied egg-rr 72.9%

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

      1. expm1-log1p-u 30.2%

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

      2. expm1-udef 24.3%

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

   7. Applied egg-rr 19.0%

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

      1. expm1-def 22.1%

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

      2. expm1-log1p 60.3%

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

      3. +-commutative 60.3%

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

      4. fma-def 60.3%

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

      5. associate-/r/ 65.0%

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

      6. *-commutative 65.0%

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

      7. associate-*r/ 66.0%

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

      8. associate-/l* 64.9%

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

      9. *-commutative 64.9%

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

      10. associate-/l* 64.9%

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

   9. Simplified 64.9%

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

   if 6.5000000000000002e-200 < l < 6.1999999999999995e132

   1. Initial program 71.1%

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

   3. Simplified 72.9%

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

      1. associate-*r/ 76.2%

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

   5. Applied egg-rr 74.5%

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

      1. add-cbrt-cube 67.9%

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

      2. add-sqr-sqrt 67.7%

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

      3. cbrt-prod 74.3%

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

   7. Applied egg-rr 74.3%

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

   9. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
   10. Step-by-step derivation

       1. *-rgt-identity 68.8%

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

       2. distribute-lft-out 82.6%

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

       3. *-commutative 82.6%

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

       4. associate-*l* 82.6%

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

       5. associate-*r* 82.6%

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

       6. metadata-eval 82.6%

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

   11. Simplified 82.6%

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

   if 6.1999999999999995e132 < l

   1. Initial program 38.4%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in h around 0 41.5%

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

      1. *-rgt-identity 41.5%

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

      2. sqrt-prod 39.3%

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

      3. expm1-log1p-u 37.4%

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

      4. expm1-udef 22.7%

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

      5. frac-times 13.0%

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

      6. sqrt-div 13.0%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 46.5%

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

   8. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. sqrt-prod 64.9%

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

   10. Applied egg-rr 64.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(-0.5, h \cdot \frac{{\left(\frac{M}{\frac{\frac{d}{D}}{0.5}}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 9: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1900000000000:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(-0.5, \frac{h \cdot {\left(\left(M \cdot D_m\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -1900000000000.0)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (if (<= l 6.5e-200)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (fma -0.5 (/ (* h (pow (* (* M D_m) (/ 0.5 d)) 2.0)) l) 1.0))
     (if (<= l 1.6e+137)
       (*
        (/ d (sqrt (* h l)))
        (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D_m d)) 2.0)))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -1900000000000.0) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else if (l <= 6.5e-200) {
		tmp = sqrt(((d / l) * (d / h))) * fma(-0.5, ((h * pow(((M * D_m) * (0.5 / d)), 2.0)) / l), 1.0);
	} else if (l <= 1.6e+137) {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.125 * ((h / l) * pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -1900000000000.0)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (l <= 6.5e-200)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(-0.5, Float64(Float64(h * (Float64(Float64(M * D_m) * Float64(0.5 / d)) ^ 2.0)) / l), 1.0));
	elseif (l <= 1.6e+137)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -1900000000000.0], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e-200], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 * N[(N[(h * N[Power[N[(N[(M * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e+137], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.9e12

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in h around 0 39.7%

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

      1. frac-2neg 39.7%

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

      2. sqrt-div 53.3%

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

   6. Applied egg-rr 53.3%

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

   if -1.9e12 < l < 6.5000000000000002e-200

   1. Initial program 68.4%

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

   3. Simplified 66.3%

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

      1. associate-*r/ 71.9%

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

   5. Applied egg-rr 72.9%

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

      1. expm1-log1p-u 30.2%

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

      2. expm1-udef 24.3%

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

   7. Applied egg-rr 19.0%

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

      1. expm1-def 22.1%

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

      2. expm1-log1p 60.3%

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

      3. +-commutative 60.3%

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

      4. fma-def 60.3%

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

      5. associate-*r/ 61.4%

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

      6. associate-*l/ 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-*r/ 59.2%

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

      9. *-commutative 59.2%

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

      10. associate-/l* 64.8%

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

      11. associate-*r/ 59.2%

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

   9. Simplified 59.2%

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

       1. associate-*r/ 64.8%

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

       2. associate-*r* 67.0%

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

   11. Applied egg-rr 67.0%

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

   if 6.5000000000000002e-200 < l < 1.60000000000000009e137

   1. Initial program 71.1%

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

   3. Simplified 72.9%

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

      1. associate-*r/ 76.2%

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

   5. Applied egg-rr 74.5%

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

      1. add-cbrt-cube 67.9%

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

      2. add-sqr-sqrt 67.7%

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

      3. cbrt-prod 74.3%

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

   7. Applied egg-rr 74.3%

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

   9. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
   10. Step-by-step derivation

       1. *-rgt-identity 68.8%

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

       2. distribute-lft-out 82.6%

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

       3. *-commutative 82.6%

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

       4. associate-*l* 82.6%

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

       5. associate-*r* 82.6%

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

       6. metadata-eval 82.6%

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

   11. Simplified 82.6%

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

   if 1.60000000000000009e137 < l

   1. Initial program 38.4%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in h around 0 41.5%

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

      1. *-rgt-identity 41.5%

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

      2. sqrt-prod 39.3%

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

      3. expm1-log1p-u 37.4%

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

      4. expm1-udef 22.7%

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

      5. frac-times 13.0%

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

      6. sqrt-div 13.0%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 46.5%

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

   8. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. sqrt-prod 64.9%

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

   10. Applied egg-rr 64.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1900000000000:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(-0.5, \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 10: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(-0.5, \frac{h \cdot {\left(\left(M \cdot D_m\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D_m}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -3.2e+14)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (if (<= l 2.7e-304)
     (*
      (sqrt (* (/ d l) (/ d h)))
      (fma -0.5 (/ (* h (pow (* (* M D_m) (/ 0.5 d)) 2.0)) l) 1.0))
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* M (* (/ D_m d) 0.5)) 2.0))))
      (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -3.2e+14) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else if (l <= 2.7e-304) {
		tmp = sqrt(((d / l) * (d / h))) * fma(-0.5, ((h * pow(((M * D_m) * (0.5 / d)), 2.0)) / l), 1.0);
	} else {
		tmp = (1.0 - (0.5 * ((h / l) * pow((M * ((D_m / d) * 0.5)), 2.0)))) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -3.2e+14)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (l <= 2.7e-304)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * fma(-0.5, Float64(Float64(h * (Float64(Float64(M * D_m) * Float64(0.5 / d)) ^ 2.0)) / l), 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D_m / d) * 0.5)) ^ 2.0)))) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -3.2e+14], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.7e-304], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 * N[(N[(h * N[Power[N[(N[(M * D$95$m), $MachinePrecision] * N[(0.5 / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.2e14

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in h around 0 39.7%

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

      1. frac-2neg 39.7%

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

      2. sqrt-div 53.3%

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

   6. Applied egg-rr 53.3%

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

   if -3.2e14 < l < 2.7000000000000001e-304

   1. Initial program 68.3%

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

   3. Simplified 68.4%

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

      1. associate-*r/ 74.5%

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

   5. Applied egg-rr 74.5%

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

      1. expm1-log1p-u 32.5%

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

      2. expm1-udef 25.8%

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

   7. Applied egg-rr 18.6%

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

      1. expm1-def 22.9%

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

      2. expm1-log1p 58.6%

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

      3. +-commutative 58.6%

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

      4. fma-def 58.6%

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

      5. associate-*r/ 58.7%

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

      6. associate-*l/ 58.6%

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

      7. *-commutative 58.6%

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

      8. associate-*r/ 58.6%

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

      9. *-commutative 58.6%

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

      10. associate-/l* 64.8%

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

      11. associate-*r/ 58.6%

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

   9. Simplified 58.6%

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

       1. associate-*r/ 64.8%

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

       2. associate-*r* 64.8%

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

   11. Applied egg-rr 64.8%

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

   if 2.7000000000000001e-304 < l

   1. Initial program 60.1%

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

   3. Simplified 59.5%

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

   5. Applied egg-rr 31.3%

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

      1. expm1-def 47.4%

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

      2. expm1-log1p 78.8%

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

      3. associate-*l* 78.8%

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

      4. *-commutative 78.8%

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

      5. associate-*l* 78.8%

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

   7. Simplified 78.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \mathsf{fma}\left(-0.5, \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(\frac{D}{d} \cdot 0.5\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 11: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{d}{-t_0}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D_m \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{t_0} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -1e+196)
     (* (sqrt (/ d h)) (/ (sqrt (- d)) (sqrt (- l))))
     (if (<= l -5.5e+58)
       (/ d (- t_0))
       (if (<= l 5.5e-303)
         (*
          (sqrt (/ (* d (/ d h)) l))
          (+ 1.0 (* -0.5 (* h (/ (pow (/ (* D_m 0.5) (/ d M)) 2.0) l)))))
         (if (<= l 2.4e+132)
           (*
            (/ d t_0)
            (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D_m d)) 2.0)))))
           (/ d (* (sqrt h) (sqrt l)))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -1e+196) {
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	} else if (l <= -5.5e+58) {
		tmp = d / -t_0;
	} else if (l <= 5.5e-303) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (l <= 2.4e+132) {
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (l <= (-1d+196)) then
        tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l))
    else if (l <= (-5.5d+58)) then
        tmp = d / -t_0
    else if (l <= 5.5d-303) then
        tmp = sqrt(((d * (d / h)) / l)) * (1.0d0 + ((-0.5d0) * (h * ((((d_m * 0.5d0) / (d / m)) ** 2.0d0) / l))))
    else if (l <= 2.4d+132) then
        tmp = (d / t_0) * (1.0d0 + ((-0.125d0) * ((h / l) * ((m * (d_m / d)) ** 2.0d0))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (l <= -1e+196) {
		tmp = Math.sqrt((d / h)) * (Math.sqrt(-d) / Math.sqrt(-l));
	} else if (l <= -5.5e+58) {
		tmp = d / -t_0;
	} else if (l <= 5.5e-303) {
		tmp = Math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (Math.pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (l <= 2.4e+132) {
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * Math.pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if l <= -1e+196:
		tmp = math.sqrt((d / h)) * (math.sqrt(-d) / math.sqrt(-l))
	elif l <= -5.5e+58:
		tmp = d / -t_0
	elif l <= 5.5e-303:
		tmp = math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (math.pow(((D_m * 0.5) / (d / M)), 2.0) / l))))
	elif l <= 2.4e+132:
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * math.pow((M * (D_m / d)), 2.0))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -1e+196)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))));
	elseif (l <= -5.5e+58)
		tmp = Float64(d / Float64(-t_0));
	elseif (l <= 5.5e-303)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(D_m * 0.5) / Float64(d / M)) ^ 2.0) / l)))));
	elseif (l <= 2.4e+132)
		tmp = Float64(Float64(d / t_0) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (l <= -1e+196)
		tmp = sqrt((d / h)) * (sqrt(-d) / sqrt(-l));
	elseif (l <= -5.5e+58)
		tmp = d / -t_0;
	elseif (l <= 5.5e-303)
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * ((((D_m * 0.5) / (d / M)) ^ 2.0) / l))));
	elseif (l <= 2.4e+132)
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * ((M * (D_m / d)) ^ 2.0))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e+196], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.5e+58], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[l, 5.5e-303], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4e+132], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -9.9999999999999995e195

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in h around 0 52.1%

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

      1. frac-2neg 56.8%

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

      2. sqrt-div 75.7%

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

   6. Applied egg-rr 65.8%

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

   if -9.9999999999999995e195 < l < -5.4999999999999999e58

   1. Initial program 44.8%

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

   3. Simplified 44.7%

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

      1. add-sqr-sqrt 44.7%

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

      2. pow2 44.7%

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

   5. Applied egg-rr 47.7%

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

      1. associate-*r/ 50.4%

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

      2. associate-*l* 50.4%

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

      3. *-commutative 50.4%

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

      4. associate-*l/ 47.6%

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

      5. *-commutative 47.6%

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

      6. associate-*l* 47.6%

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

      7. associate-*l* 47.6%

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

      8. metadata-eval 47.6%

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

      9. associate-/r/ 47.6%

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

      10. associate-/l* 47.5%

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

      11. associate-/r/ 47.6%

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

      12. *-commutative 47.6%

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

      13. associate-/r* 47.6%

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

      14. metadata-eval 47.6%

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

   7. Simplified 47.6%

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

      1. clear-num 47.6%

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

      2. sqrt-div 47.6%

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

      3. metadata-eval 47.6%

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

   9. Applied egg-rr 47.6%

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

   10. Taylor expanded in d around -inf 53.8%

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

       1. mul-1-neg 53.8%

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

       2. unpow-1 53.8%

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

       3. metadata-eval 53.8%

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

       4. pow-sqr 53.8%

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

       5. rem-sqrt-square 53.8%

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

       6. rem-cube-cbrt 53.1%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 52.9%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 52.9%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 53.1%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 53.8%

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

       11. *-commutative 53.8%

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

   12. Simplified 53.8%

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

   if -5.4999999999999999e58 < l < 5.50000000000000018e-303

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. expm1-log1p-u 32.2%

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

      2. expm1-udef 23.1%

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

   7. Applied egg-rr 16.0%

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

   9. Simplified 56.0%

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

   if 5.50000000000000018e-303 < l < 2.4000000000000001e132

   1. Initial program 70.4%

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

   3. Simplified 69.3%

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

      1. associate-*r/ 72.9%

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

   5. Applied egg-rr 72.8%

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

      1. add-cbrt-cube 65.8%

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

      2. add-sqr-sqrt 65.7%

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

      3. cbrt-prod 72.6%

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

   7. Applied egg-rr 72.6%

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

   9. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
   10. Step-by-step derivation

       1. *-rgt-identity 57.7%

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

       2. distribute-lft-out 78.4%

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

       3. *-commutative 78.4%

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

       4. associate-*l* 78.4%

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

       5. associate-*r* 78.4%

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

       6. metadata-eval 78.4%

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

   11. Simplified 78.4%

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

   if 2.4000000000000001e132 < l

   1. Initial program 38.4%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in h around 0 41.5%

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

      1. *-rgt-identity 41.5%

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

      2. sqrt-prod 39.3%

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

      3. expm1-log1p-u 37.4%

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

      4. expm1-udef 22.7%

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

      5. frac-times 13.0%

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

      6. sqrt-div 13.0%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 46.5%

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

   8. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. sqrt-prod 64.9%

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

   10. Applied egg-rr 64.9%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 12: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D_m \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= l -2.1e+15)
   (* (/ (sqrt (- d)) (sqrt (- h))) (sqrt (/ d l)))
   (if (<= l 1.3e-300)
     (*
      (sqrt (/ (* d (/ d h)) l))
      (+ 1.0 (* -0.5 (* h (/ (pow (/ (* D_m 0.5) (/ d M)) 2.0) l)))))
     (if (<= l 4.2e+130)
       (*
        (/ d (sqrt (* h l)))
        (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D_m d)) 2.0)))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -2.1e+15) {
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	} else if (l <= 1.3e-300) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (l <= 4.2e+130) {
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.125 * ((h / l) * pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (l <= (-2.1d+15)) then
        tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l))
    else if (l <= 1.3d-300) then
        tmp = sqrt(((d * (d / h)) / l)) * (1.0d0 + ((-0.5d0) * (h * ((((d_m * 0.5d0) / (d / m)) ** 2.0d0) / l))))
    else if (l <= 4.2d+130) then
        tmp = (d / sqrt((h * l))) * (1.0d0 + ((-0.125d0) * ((h / l) * ((m * (d_m / d)) ** 2.0d0))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (l <= -2.1e+15) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * Math.sqrt((d / l));
	} else if (l <= 1.3e-300) {
		tmp = Math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (Math.pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (l <= 4.2e+130) {
		tmp = (d / Math.sqrt((h * l))) * (1.0 + (-0.125 * ((h / l) * Math.pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if l <= -2.1e+15:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * math.sqrt((d / l))
	elif l <= 1.3e-300:
		tmp = math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (math.pow(((D_m * 0.5) / (d / M)), 2.0) / l))))
	elif l <= 4.2e+130:
		tmp = (d / math.sqrt((h * l))) * (1.0 + (-0.125 * ((h / l) * math.pow((M * (D_m / d)), 2.0))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (l <= -2.1e+15)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	elseif (l <= 1.3e-300)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(D_m * 0.5) / Float64(d / M)) ^ 2.0) / l)))));
	elseif (l <= 4.2e+130)
		tmp = Float64(Float64(d / sqrt(Float64(h * l))) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (l <= -2.1e+15)
		tmp = (sqrt(-d) / sqrt(-h)) * sqrt((d / l));
	elseif (l <= 1.3e-300)
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * ((((D_m * 0.5) / (d / M)) ^ 2.0) / l))));
	elseif (l <= 4.2e+130)
		tmp = (d / sqrt((h * l))) * (1.0 + (-0.125 * ((h / l) * ((M * (D_m / d)) ^ 2.0))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[l, -2.1e+15], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e-300], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e+130], N[(N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.1e15

   1. Initial program 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in h around 0 39.7%

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

      1. frac-2neg 39.7%

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

      2. sqrt-div 53.3%

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

   6. Applied egg-rr 53.3%

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

   if -2.1e15 < l < 1.29999999999999998e-300

   1. Initial program 68.3%

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

   3. Simplified 68.4%

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

      1. associate-*r/ 74.5%

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

   5. Applied egg-rr 74.5%

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

      1. expm1-log1p-u 32.5%

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

      2. expm1-udef 25.8%

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

   7. Applied egg-rr 18.6%

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

   9. Simplified 60.3%

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

   if 1.29999999999999998e-300 < l < 4.19999999999999981e130

   1. Initial program 70.4%

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

   3. Simplified 69.3%

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

      1. associate-*r/ 72.9%

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

   5. Applied egg-rr 72.8%

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

      1. add-cbrt-cube 65.8%

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

      2. add-sqr-sqrt 65.7%

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

      3. cbrt-prod 72.6%

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

   7. Applied egg-rr 72.6%

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

   9. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
   10. Step-by-step derivation

       1. *-rgt-identity 57.7%

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

       2. distribute-lft-out 78.4%

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

       3. *-commutative 78.4%

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

       4. associate-*l* 78.4%

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

       5. associate-*r* 78.4%

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

       6. metadata-eval 78.4%

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

   11. Simplified 78.4%

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

   if 4.19999999999999981e130 < l

   1. Initial program 38.4%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in h around 0 41.5%

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

      1. *-rgt-identity 41.5%

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

      2. sqrt-prod 39.3%

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

      3. expm1-log1p-u 37.4%

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

      4. expm1-udef 22.7%

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

      5. frac-times 13.0%

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

      6. sqrt-div 13.0%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 46.5%

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

   8. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. sqrt-prod 64.9%

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

   10. Applied egg-rr 64.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 13: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{d}{-t_0}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D_m \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{t_0} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -4.8e+58)
     (/ d (- t_0))
     (if (<= l 5.4e-305)
       (*
        (sqrt (/ (* d (/ d h)) l))
        (+ 1.0 (* -0.5 (* h (/ (pow (/ (* D_m 0.5) (/ d M)) 2.0) l)))))
       (if (<= l 4.6e+127)
         (* (/ d t_0) (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D_m d)) 2.0)))))
         (/ d (* (sqrt h) (sqrt l))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -4.8e+58) {
		tmp = d / -t_0;
	} else if (l <= 5.4e-305) {
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (l <= 4.6e+127) {
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (l <= (-4.8d+58)) then
        tmp = d / -t_0
    else if (l <= 5.4d-305) then
        tmp = sqrt(((d * (d / h)) / l)) * (1.0d0 + ((-0.5d0) * (h * ((((d_m * 0.5d0) / (d / m)) ** 2.0d0) / l))))
    else if (l <= 4.6d+127) then
        tmp = (d / t_0) * (1.0d0 + ((-0.125d0) * ((h / l) * ((m * (d_m / d)) ** 2.0d0))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (l <= -4.8e+58) {
		tmp = d / -t_0;
	} else if (l <= 5.4e-305) {
		tmp = Math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (Math.pow(((D_m * 0.5) / (d / M)), 2.0) / l))));
	} else if (l <= 4.6e+127) {
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * Math.pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if l <= -4.8e+58:
		tmp = d / -t_0
	elif l <= 5.4e-305:
		tmp = math.sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * (math.pow(((D_m * 0.5) / (d / M)), 2.0) / l))))
	elif l <= 4.6e+127:
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * math.pow((M * (D_m / d)), 2.0))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -4.8e+58)
		tmp = Float64(d / Float64(-t_0));
	elseif (l <= 5.4e-305)
		tmp = Float64(sqrt(Float64(Float64(d * Float64(d / h)) / l)) * Float64(1.0 + Float64(-0.5 * Float64(h * Float64((Float64(Float64(D_m * 0.5) / Float64(d / M)) ^ 2.0) / l)))));
	elseif (l <= 4.6e+127)
		tmp = Float64(Float64(d / t_0) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (l <= -4.8e+58)
		tmp = d / -t_0;
	elseif (l <= 5.4e-305)
		tmp = sqrt(((d * (d / h)) / l)) * (1.0 + (-0.5 * (h * ((((D_m * 0.5) / (d / M)) ^ 2.0) / l))));
	elseif (l <= 4.6e+127)
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * ((M * (D_m / d)) ^ 2.0))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.8e+58], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[l, 5.4e-305], N[(N[Sqrt[N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(h * N[(N[Power[N[(N[(D$95$m * 0.5), $MachinePrecision] / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.6e+127], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.8e58

   1. Initial program 49.1%

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

   3. Simplified 49.1%

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

      1. add-sqr-sqrt 49.1%

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

      2. pow2 49.1%

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

   5. Applied egg-rr 51.0%

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

      1. associate-*r/ 52.7%

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

      2. associate-*l* 52.7%

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

      3. *-commutative 52.7%

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

      4. associate-*l/ 50.9%

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

      5. *-commutative 50.9%

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

      6. associate-*l* 50.9%

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

      7. associate-*l* 51.0%

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

      8. metadata-eval 51.0%

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

      9. associate-/r/ 51.0%

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

      10. associate-/l* 50.9%

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

      11. associate-/r/ 51.0%

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

      12. *-commutative 51.0%

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

      13. associate-/r* 51.0%

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

      14. metadata-eval 51.0%

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

   7. Simplified 51.0%

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

      1. clear-num 51.0%

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

      2. sqrt-div 50.9%

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

      3. metadata-eval 50.9%

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

   9. Applied egg-rr 50.9%

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

   10. Taylor expanded in d around -inf 47.7%

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

       1. mul-1-neg 47.7%

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

       2. unpow-1 47.7%

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

       3. metadata-eval 47.7%

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

       4. pow-sqr 47.7%

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

       5. rem-sqrt-square 47.7%

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

       6. rem-cube-cbrt 47.1%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 47.0%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 47.0%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 47.1%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 47.7%

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

       11. *-commutative 47.7%

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

   12. Simplified 47.7%

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

   if -4.8e58 < l < 5.3999999999999998e-305

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. associate-*r/ 72.5%

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

   5. Applied egg-rr 72.5%

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

      1. expm1-log1p-u 32.2%

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

      2. expm1-udef 23.1%

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

   7. Applied egg-rr 16.0%

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

   9. Simplified 56.0%

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

   if 5.3999999999999998e-305 < l < 4.6000000000000003e127

   1. Initial program 70.4%

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

   3. Simplified 69.3%

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

      1. associate-*r/ 72.9%

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

   5. Applied egg-rr 72.8%

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

      1. add-cbrt-cube 65.8%

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

      2. add-sqr-sqrt 65.7%

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

      3. cbrt-prod 72.6%

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

   7. Applied egg-rr 72.6%

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

   9. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
   10. Step-by-step derivation

       1. *-rgt-identity 57.7%

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

       2. distribute-lft-out 78.4%

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

       3. *-commutative 78.4%

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

       4. associate-*l* 78.4%

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

       5. associate-*r* 78.4%

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

       6. metadata-eval 78.4%

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

   11. Simplified 78.4%

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

   if 4.6000000000000003e127 < l

   1. Initial program 38.4%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in h around 0 41.5%

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

      1. *-rgt-identity 41.5%

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

      2. sqrt-prod 39.3%

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

      3. expm1-log1p-u 37.4%

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

      4. expm1-udef 22.7%

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

      5. frac-times 13.0%

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

      6. sqrt-div 13.0%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 46.5%

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

   8. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. sqrt-prod 64.9%

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

   10. Applied egg-rr 64.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{d \cdot \frac{d}{h}}{\ell}} \cdot \left(1 + -0.5 \cdot \left(h \cdot \frac{{\left(\frac{D \cdot 0.5}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 14: 57.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{-t_0}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{d}{t_0} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D_m}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -2e-310)
     (/ d (- t_0))
     (if (<= l 5.8e+130)
       (* (/ d t_0) (+ 1.0 (* -0.125 (* (/ h l) (pow (* M (/ D_m d)) 2.0)))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -2e-310) {
		tmp = d / -t_0;
	} else if (l <= 5.8e+130) {
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (l <= (-2d-310)) then
        tmp = d / -t_0
    else if (l <= 5.8d+130) then
        tmp = (d / t_0) * (1.0d0 + ((-0.125d0) * ((h / l) * ((m * (d_m / d)) ** 2.0d0))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (l <= -2e-310) {
		tmp = d / -t_0;
	} else if (l <= 5.8e+130) {
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * Math.pow((M * (D_m / d)), 2.0))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if l <= -2e-310:
		tmp = d / -t_0
	elif l <= 5.8e+130:
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * math.pow((M * (D_m / d)), 2.0))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(d / Float64(-t_0));
	elseif (l <= 5.8e+130)
		tmp = Float64(Float64(d / t_0) * Float64(1.0 + Float64(-0.125 * Float64(Float64(h / l) * (Float64(M * Float64(D_m / d)) ^ 2.0)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = d / -t_0;
	elseif (l <= 5.8e+130)
		tmp = (d / t_0) * (1.0 + (-0.125 * ((h / l) * ((M * (D_m / d)) ^ 2.0))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(d / (-t$95$0)), $MachinePrecision], If[LessEqual[l, 5.8e+130], N[(N[(d / t$95$0), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 59.7%

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

   3. Simplified 59.7%

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

      1. add-sqr-sqrt 59.7%

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

      2. pow2 59.7%

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

   5. Applied egg-rr 61.3%

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

      1. associate-*r/ 62.6%

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

      2. associate-*l* 62.6%

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

      3. *-commutative 62.6%

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

      4. associate-*l/ 61.9%

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

      5. *-commutative 61.9%

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

      6. associate-*l* 61.9%

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

      7. associate-*l* 61.2%

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

      8. metadata-eval 61.2%

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

      9. associate-/r/ 61.2%

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

      10. associate-/l* 61.2%

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

      11. associate-/r/ 61.2%

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

      12. *-commutative 61.2%

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

      13. associate-/r* 61.2%

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

      14. metadata-eval 61.2%

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

   7. Simplified 61.2%

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

      1. clear-num 61.2%

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

      2. sqrt-div 61.2%

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

      3. metadata-eval 61.2%

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

   9. Applied egg-rr 61.2%

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

   10. Taylor expanded in d around -inf 38.9%

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

       1. mul-1-neg 38.9%

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

       2. unpow-1 38.9%

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

       3. metadata-eval 38.9%

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

       4. pow-sqr 38.9%

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

       5. rem-sqrt-square 38.9%

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

       6. rem-cube-cbrt 38.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 38.4%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 38.4%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 38.5%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 38.9%

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

       11. *-commutative 38.9%

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

   12. Simplified 38.9%

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

   if -1.999999999999994e-310 < l < 5.7999999999999998e130

   1. Initial program 70.2%

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

   3. Simplified 69.2%

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

      1. associate-*r/ 72.7%

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

   5. Applied egg-rr 72.6%

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

      1. add-cbrt-cube 64.7%

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

      2. add-sqr-sqrt 64.6%

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

      3. cbrt-prod 72.5%

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

   7. Applied egg-rr 72.5%

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

   9. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(0.25 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \cdot -0.5\right)} \]
   10. Step-by-step derivation

       1. *-rgt-identity 56.8%

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

       2. distribute-lft-out 78.0%

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

       3. *-commutative 78.0%

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

       4. associate-*l* 78.0%

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

       5. associate-*r* 78.0%

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

       6. metadata-eval 78.0%

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

   11. Simplified 78.0%

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

   if 5.7999999999999998e130 < l

   1. Initial program 38.4%

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

   3. Simplified 38.6%

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

   4. Taylor expanded in h around 0 41.5%

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

      1. *-rgt-identity 41.5%

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

      2. sqrt-prod 39.3%

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

      3. expm1-log1p-u 37.4%

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

      4. expm1-udef 22.7%

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

      5. frac-times 13.0%

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

      6. sqrt-div 13.0%

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

      7. sqrt-unprod 28.6%

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

      8. add-sqr-sqrt 28.6%

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

   6. Applied egg-rr 28.6%

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

      1. expm1-def 43.5%

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

      2. expm1-log1p 46.5%

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

   8. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. sqrt-prod 64.9%

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

   10. Applied egg-rr 64.9%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 5.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + -0.125 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 15: 46.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.8 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -2.8e+212)
   (sqrt (/ (/ d h) (/ l d)))
   (if (<= h 9.5e-303) (/ d (- (sqrt (* h l)))) (/ d (* (sqrt h) (sqrt l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -2.8e+212) {
		tmp = sqrt(((d / h) / (l / d)));
	} else if (h <= 9.5e-303) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-2.8d+212)) then
        tmp = sqrt(((d / h) / (l / d)))
    else if (h <= 9.5d-303) then
        tmp = d / -sqrt((h * l))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -2.8e+212) {
		tmp = Math.sqrt(((d / h) / (l / d)));
	} else if (h <= 9.5e-303) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if h <= -2.8e+212:
		tmp = math.sqrt(((d / h) / (l / d)))
	elif h <= 9.5e-303:
		tmp = d / -math.sqrt((h * l))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -2.8e+212)
		tmp = sqrt(Float64(Float64(d / h) / Float64(l / d)));
	elseif (h <= 9.5e-303)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (h <= -2.8e+212)
		tmp = sqrt(((d / h) / (l / d)));
	elseif (h <= 9.5e-303)
		tmp = d / -sqrt((h * l));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -2.8e+212], N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[h, 9.5e-303], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.79999999999999997e212

   1. Initial program 48.7%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in h around 0 35.5%

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

      1. *-rgt-identity 35.5%

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

      2. sqrt-prod 35.5%

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

      3. expm1-log1p-u 34.9%

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

      4. expm1-udef 15.2%

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

      5. frac-times 11.4%

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

      6. sqrt-div 11.4%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. expm1-def 5.4%

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

      2. expm1-log1p 6.1%

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

   8. Simplified 6.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 15.0%

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

      3. sqrt-div 15.0%

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

      4. frac-times 35.5%

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

      5. sqrt-prod 35.5%

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

      6. clear-num 35.5%

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

      7. sqrt-div 37.8%

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

      8. metadata-eval 37.8%

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

      9. un-div-inv 37.8%

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

      10. sqrt-undiv 37.8%

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

   10. Applied egg-rr 37.8%

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

   if -2.79999999999999997e212 < h < 9.4999999999999999e-303

   1. Initial program 63.2%

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

   3. Simplified 63.3%

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

      1. add-sqr-sqrt 63.3%

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

      2. pow2 63.3%

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

   5. Applied egg-rr 65.2%

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

      1. associate-*r/ 66.9%

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

      2. associate-*l* 66.9%

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

      3. *-commutative 66.9%

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

      4. associate-*l/ 66.1%

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

      5. *-commutative 66.1%

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

      6. associate-*l* 66.1%

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

      7. associate-*l* 65.2%

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

      8. metadata-eval 65.2%

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

      9. associate-/r/ 65.2%

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

      10. associate-/l* 65.1%

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

      11. associate-/r/ 65.1%

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

      12. *-commutative 65.1%

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

      13. associate-/r* 65.1%

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

      14. metadata-eval 65.1%

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

   7. Simplified 65.1%

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

      1. clear-num 65.1%

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

      2. sqrt-div 65.2%

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

      3. metadata-eval 65.2%

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

   9. Applied egg-rr 65.2%

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

   10. Taylor expanded in d around -inf 44.9%

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

       1. mul-1-neg 44.9%

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

       2. unpow-1 44.9%

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

       3. metadata-eval 44.9%

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

       4. pow-sqr 45.0%

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

       5. rem-sqrt-square 45.0%

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

       6. rem-cube-cbrt 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 44.5%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 44.5%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 45.0%

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

       11. *-commutative 45.0%

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

   12. Simplified 45.0%

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

   if 9.4999999999999999e-303 < h

   1. Initial program 60.0%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in h around 0 38.8%

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

      1. *-rgt-identity 38.8%

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

      2. sqrt-prod 32.8%

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

      3. expm1-log1p-u 31.8%

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

      4. expm1-udef 21.1%

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

      5. frac-times 16.6%

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

      6. sqrt-div 20.7%

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

      7. sqrt-unprod 31.0%

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

      8. add-sqr-sqrt 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. expm1-def 42.5%

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

      2. expm1-log1p 44.9%

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

   8. Simplified 44.9%

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

      1. *-commutative 44.9%

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

      2. sqrt-prod 52.8%

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

   10. Applied egg-rr 52.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.8 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]

Alternative 16: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -1.7 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -1.7e+214)
   (sqrt (/ (/ d h) (/ l d)))
   (if (<= h 9.5e-303)
     (/ d (- (sqrt (* h l))))
     (* d (sqrt (* (/ 1.0 h) (/ 1.0 l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -1.7e+214) {
		tmp = sqrt(((d / h) / (l / d)));
	} else if (h <= 9.5e-303) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-1.7d+214)) then
        tmp = sqrt(((d / h) / (l / d)))
    else if (h <= 9.5d-303) then
        tmp = d / -sqrt((h * l))
    else
        tmp = d * sqrt(((1.0d0 / h) * (1.0d0 / l)))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -1.7e+214) {
		tmp = Math.sqrt(((d / h) / (l / d)));
	} else if (h <= 9.5e-303) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) * (1.0 / l)));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if h <= -1.7e+214:
		tmp = math.sqrt(((d / h) / (l / d)))
	elif h <= 9.5e-303:
		tmp = d / -math.sqrt((h * l))
	else:
		tmp = d * math.sqrt(((1.0 / h) * (1.0 / l)))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -1.7e+214)
		tmp = sqrt(Float64(Float64(d / h) / Float64(l / d)));
	elseif (h <= 9.5e-303)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) * Float64(1.0 / l))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (h <= -1.7e+214)
		tmp = sqrt(((d / h) / (l / d)));
	elseif (h <= 9.5e-303)
		tmp = d / -sqrt((h * l));
	else
		tmp = d * sqrt(((1.0 / h) * (1.0 / l)));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -1.7e+214], N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[h, 9.5e-303], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.6999999999999999e214

   1. Initial program 48.7%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in h around 0 35.5%

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

      1. *-rgt-identity 35.5%

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

      2. sqrt-prod 35.5%

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

      3. expm1-log1p-u 34.9%

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

      4. expm1-udef 15.2%

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

      5. frac-times 11.4%

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

      6. sqrt-div 11.4%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. expm1-def 5.4%

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

      2. expm1-log1p 6.1%

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

   8. Simplified 6.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 15.0%

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

      3. sqrt-div 15.0%

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

      4. frac-times 35.5%

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

      5. sqrt-prod 35.5%

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

      6. clear-num 35.5%

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

      7. sqrt-div 37.8%

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

      8. metadata-eval 37.8%

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

      9. un-div-inv 37.8%

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

      10. sqrt-undiv 37.8%

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

   10. Applied egg-rr 37.8%

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

   if -1.6999999999999999e214 < h < 9.4999999999999999e-303

   1. Initial program 63.2%

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

   3. Simplified 63.3%

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

      1. add-sqr-sqrt 63.3%

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

      2. pow2 63.3%

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

   5. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\left(M \cdot 0.5\right) \cdot D}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. associate-*l* 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. *-commutative 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot \color{blue}{\left(D \cdot 0.5\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. associate-*l/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{1}{2}} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/r/ 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{2}{\frac{M}{d}}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/l* 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. associate-/r/ 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. associate-/r* 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{\color{blue}{0.5}}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   7. Simplified 65.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   8. Step-by-step derivation

      1. clear-num 65.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   9. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Taylor expanded in d around -inf 44.9%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 44.9%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. unpow-1 44.9%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. metadata-eval 44.9%

          \[\leadsto -d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       4. pow-sqr 45.0%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       5. rem-sqrt-square 45.0%

          \[\leadsto -d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       6. rem-cube-cbrt 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 44.5%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 44.5%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 45.0%

           \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       11. *-commutative 45.0%

           \[\leadsto -\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot d} \]

   12. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 9.4999999999999999e-303 < h

   1. Initial program 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 44.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. inv-pow 44.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

      2. *-commutative 44.8%

         \[\leadsto d \cdot \sqrt{{\color{blue}{\left(\ell \cdot h\right)}}^{-1}} \]

      3. unpow-prod-down 45.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{{\ell}^{-1} \cdot {h}^{-1}}} \]

      4. inv-pow 45.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell}} \cdot {h}^{-1}} \]

      5. inv-pow 45.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\ell} \cdot \color{blue}{\frac{1}{h}}} \]

   6. Applied egg-rr 45.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{\ell} \cdot \frac{1}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.7 \cdot 10^{+214}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h} \cdot \frac{1}{\ell}}\\ \end{array} \]

Alternative 17: 42.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{-302}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= h -8e+212)
   (sqrt (/ (/ d h) (/ l d)))
   (if (<= h 1.05e-302)
     (/ d (- (sqrt (* h l))))
     (* d (sqrt (/ (/ 1.0 h) l))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -8e+212) {
		tmp = sqrt(((d / h) / (l / d)));
	} else if (h <= 1.05e-302) {
		tmp = d / -sqrt((h * l));
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (h <= (-8d+212)) then
        tmp = sqrt(((d / h) / (l / d)))
    else if (h <= 1.05d-302) then
        tmp = d / -sqrt((h * l))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (h <= -8e+212) {
		tmp = Math.sqrt(((d / h) / (l / d)));
	} else if (h <= 1.05e-302) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if h <= -8e+212:
		tmp = math.sqrt(((d / h) / (l / d)))
	elif h <= 1.05e-302:
		tmp = d / -math.sqrt((h * l))
	else:
		tmp = d * math.sqrt(((1.0 / h) / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (h <= -8e+212)
		tmp = sqrt(Float64(Float64(d / h) / Float64(l / d)));
	elseif (h <= 1.05e-302)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (h <= -8e+212)
		tmp = sqrt(((d / h) / (l / d)));
	elseif (h <= 1.05e-302)
		tmp = d / -sqrt((h * l));
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[h, -8e+212], N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[h, 1.05e-302], N[(d / (-N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.9999999999999993e212

   1. Initial program 48.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 35.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 35.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-prod 35.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      3. expm1-log1p-u 34.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      4. expm1-udef 15.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]

      5. frac-times 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      6. sqrt-div 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      7. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      8. add-sqr-sqrt 5.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   6. Applied egg-rr 5.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 5.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 6.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

      2. sqrt-unprod 15.0%

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-div 15.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      4. frac-times 35.5%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      5. sqrt-prod 35.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

      6. clear-num 35.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}} \]

      7. sqrt-div 37.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}} \]

      8. metadata-eval 37.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}} \]

      9. un-div-inv 37.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \]

      10. sqrt-undiv 37.8%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   10. Applied egg-rr 37.8%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   if -7.9999999999999993e212 < h < 1.05000000000000006e-302

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   5. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\left(M \cdot 0.5\right) \cdot D}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. associate-*l* 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. *-commutative 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot \color{blue}{\left(D \cdot 0.5\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. associate-*l/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{1}{2}} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/r/ 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{2}{\frac{M}{d}}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/l* 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. associate-/r/ 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. associate-/r* 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{\color{blue}{0.5}}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   7. Simplified 65.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   8. Step-by-step derivation

      1. clear-num 65.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   9. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Taylor expanded in d around -inf 44.9%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 44.9%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. unpow-1 44.9%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. metadata-eval 44.9%

          \[\leadsto -d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       4. pow-sqr 45.0%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       5. rem-sqrt-square 45.0%

          \[\leadsto -d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       6. rem-cube-cbrt 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 44.5%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 44.5%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 45.0%

           \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       11. *-commutative 45.0%

           \[\leadsto -\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot d} \]

   12. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 1.05000000000000006e-302 < h

   1. Initial program 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

   4. Taylor expanded in d around inf 44.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. associate-/r* 45.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   6. Simplified 45.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 1.05 \cdot 10^{-302}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]

Alternative 18: 42.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;h \leq -1.65 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{d}{-t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= h -1.65e+213)
     (sqrt (/ (/ d h) (/ l d)))
     (if (<= h 9.5e-303) (/ d (- t_0)) (/ d t_0)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (h <= -1.65e+213) {
		tmp = sqrt(((d / h) / (l / d)));
	} else if (h <= 9.5e-303) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (h <= (-1.65d+213)) then
        tmp = sqrt(((d / h) / (l / d)))
    else if (h <= 9.5d-303) then
        tmp = d / -t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (h <= -1.65e+213) {
		tmp = Math.sqrt(((d / h) / (l / d)));
	} else if (h <= 9.5e-303) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if h <= -1.65e+213:
		tmp = math.sqrt(((d / h) / (l / d)))
	elif h <= 9.5e-303:
		tmp = d / -t_0
	else:
		tmp = d / t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (h <= -1.65e+213)
		tmp = sqrt(Float64(Float64(d / h) / Float64(l / d)));
	elseif (h <= 9.5e-303)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (h <= -1.65e+213)
		tmp = sqrt(((d / h) / (l / d)));
	elseif (h <= 9.5e-303)
		tmp = d / -t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1.65e+213], N[Sqrt[N[(N[(d / h), $MachinePrecision] / N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[h, 9.5e-303], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.6500000000000001e213

   1. Initial program 48.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 35.5%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 35.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-prod 35.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      3. expm1-log1p-u 34.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      4. expm1-udef 15.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]

      5. frac-times 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      6. sqrt-div 11.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      7. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      8. add-sqr-sqrt 5.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   6. Applied egg-rr 5.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 5.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 6.1%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Simplified 6.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

      2. sqrt-unprod 15.0%

         \[\leadsto \frac{\color{blue}{\sqrt{d \cdot d}}}{\sqrt{h \cdot \ell}} \]

      3. sqrt-div 15.0%

         \[\leadsto \color{blue}{\sqrt{\frac{d \cdot d}{h \cdot \ell}}} \]

      4. frac-times 35.5%

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      5. sqrt-prod 35.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

      6. clear-num 35.5%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}} \]

      7. sqrt-div 37.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}} \]

      8. metadata-eval 37.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}} \]

      9. un-div-inv 37.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}} \]

      10. sqrt-undiv 37.8%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   10. Applied egg-rr 37.8%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \]

   if -1.6500000000000001e213 < h < 9.4999999999999999e-303

   1. Initial program 63.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 63.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   5. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\left(M \cdot 0.5\right) \cdot D}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. associate-*l* 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. *-commutative 66.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot \color{blue}{\left(D \cdot 0.5\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. associate-*l/ 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l* 66.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{1}{2}} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/r/ 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{2}{\frac{M}{d}}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/l* 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. associate-/r/ 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. associate-/r* 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 65.1%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{\color{blue}{0.5}}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   7. Simplified 65.1%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   8. Step-by-step derivation

      1. clear-num 65.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   9. Applied egg-rr 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Taylor expanded in d around -inf 44.9%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 44.9%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. unpow-1 44.9%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. metadata-eval 44.9%

          \[\leadsto -d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       4. pow-sqr 45.0%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       5. rem-sqrt-square 45.0%

          \[\leadsto -d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       6. rem-cube-cbrt 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 44.5%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 44.5%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 44.5%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 45.0%

           \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       11. *-commutative 45.0%

           \[\leadsto -\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot d} \]

   12. Simplified 45.0%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 9.4999999999999999e-303 < h

   1. Initial program 60.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 59.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 38.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 38.8%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-prod 32.8%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      3. expm1-log1p-u 31.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      4. expm1-udef 21.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]

      5. frac-times 16.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      6. sqrt-div 20.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      7. sqrt-unprod 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      8. add-sqr-sqrt 31.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   6. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 42.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 44.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Simplified 44.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.65 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}\\ \mathbf{elif}\;h \leq 9.5 \cdot 10^{-303}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 19: 42.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;d \leq 5.3 \cdot 10^{-241}:\\ \;\;\;\;\frac{d}{-t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t_0}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* h l)))) (if (<= d 5.3e-241) (/ d (- t_0)) (/ d t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (d <= 5.3e-241) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h * l))
    if (d <= 5.3d-241) then
        tmp = d / -t_0
    else
        tmp = d / t_0
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h * l));
	double tmp;
	if (d <= 5.3e-241) {
		tmp = d / -t_0;
	} else {
		tmp = d / t_0;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if d <= 5.3e-241:
		tmp = d / -t_0
	else:
		tmp = d / t_0
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (d <= 5.3e-241)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = Float64(d / t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (d <= 5.3e-241)
		tmp = d / -t_0;
	else
		tmp = d / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 5.3e-241], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 5.2999999999999998e-241

   1. Initial program 56.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 56.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{D}{2} \cdot \frac{M}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   5. Applied egg-rr 58.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{\left(M \cdot 0.5\right) \cdot D}{d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. associate-*l* 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{M \cdot \left(0.5 \cdot D\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. *-commutative 59.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot \color{blue}{\left(D \cdot 0.5\right)}}{d} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. associate-*l/ 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{d} \cdot \left(D \cdot 0.5\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. *-commutative 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\left(D \cdot 0.5\right) \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. associate-*l* 58.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \left(0.5 \cdot \frac{M}{d}\right)\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      8. metadata-eval 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{\frac{1}{2}} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      9. associate-/r/ 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\frac{1}{\frac{2}{\frac{M}{d}}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. associate-/l* 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{1}{\color{blue}{\frac{2 \cdot d}{M}}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. associate-/r/ 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{1}{2 \cdot d} \cdot M\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(M \cdot \frac{1}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. associate-/r* 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \color{blue}{\frac{\frac{1}{2}}{d}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{\color{blue}{0.5}}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   7. Simplified 58.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   8. Step-by-step derivation

      1. clear-num 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   9. Applied egg-rr 58.0%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(M \cdot \frac{0.5}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Taylor expanded in d around -inf 37.5%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 37.5%

          \[\leadsto \color{blue}{-d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

       2. unpow-1 37.5%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       3. metadata-eval 37.5%

          \[\leadsto -d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       4. pow-sqr 37.5%

          \[\leadsto -d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       5. rem-sqrt-square 36.8%

          \[\leadsto -d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       6. rem-cube-cbrt 36.4%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}}\right| \]

       7. sqr-pow 36.4%

          \[\leadsto -d \cdot \left|\color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}}\right| \]

       8. fabs-sqr 36.4%

          \[\leadsto -d \cdot \color{blue}{\left({\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{\left(\frac{3}{2}\right)}\right)} \]

       9. sqr-pow 36.4%

          \[\leadsto -d \cdot \color{blue}{{\left(\sqrt[3]{{\left(h \cdot \ell\right)}^{-0.5}}\right)}^{3}} \]

       10. rem-cube-cbrt 36.8%

           \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       11. *-commutative 36.8%

           \[\leadsto -\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot d} \]

   12. Simplified 36.8%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if 5.2999999999999998e-241 < d

   1. Initial program 64.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

   4. Taylor expanded in h around 0 42.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
   5. Step-by-step derivation

      1. *-rgt-identity 42.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

      2. sqrt-prod 35.5%

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

      3. expm1-log1p-u 34.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

      4. expm1-udef 22.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]

      5. frac-times 17.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

      6. sqrt-div 22.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      7. sqrt-unprod 33.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

      8. add-sqr-sqrt 33.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   6. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 46.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 48.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

   8. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5.3 \cdot 10^{-241}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]

Alternative 20: 26.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* h l))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((h * l));
}

Fortran

D_m = abs(D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((h * l))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((h * l));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((h * l))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((h * l));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.0%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 60.0%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{\frac{2}{M} \cdot d}\right)}^{2} \cdot -0.5, 1\right)\right)} \]

4. Taylor expanded in h around 0 37.3%

   \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{1}\right) \]
5. Step-by-step derivation

   1. *-rgt-identity 37.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

   2. sqrt-prod 31.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

   3. expm1-log1p-u 30.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)\right)} \]

   4. expm1-udef 20.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)} - 1} \]

   5. frac-times 16.4%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}}\right)} - 1 \]

   6. sqrt-div 19.8%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

   7. sqrt-unprod 15.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. add-sqr-sqrt 17.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{d}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

6. Applied egg-rr 17.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 23.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   2. expm1-log1p 26.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

8. Simplified 26.5%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

9. Final simplification 26.5%

   \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

Reproduce

herbie shell --seed 2023314 
(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)))))