Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.3% → 84.8%

Time: 25.1s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 84.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M\_m \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M\_m \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -4e-310)
     (*
      (/ t_0 (sqrt (- h)))
      (*
       (/ t_0 (sqrt (- l)))
       (- 1.0 (* h (* 0.125 (/ (pow (* M_m (/ D d)) 2.0) l))))))
     (*
      d
      (/
       (fma (/ h l) (* -0.5 (pow (* D (/ (* M_m 0.5) d)) 2.0)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -4e-310) {
		tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0 - (h * (0.125 * (pow((M_m * (D / d)), 2.0) / l)))));
	} else {
		tmp = d * (fma((h / l), (-0.5 * pow((D * ((M_m * 0.5) / d)), 2.0)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M_m * Float64(D / d)) ^ 2.0) / l))))));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(Float64(M_m * 0.5) / d)) ^ 2.0)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 62.8%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in h around -inf 37.8%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-1 \cdot \left(h \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 37.8%

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

      2. neg-mul-1 37.8%

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

      3. sub-neg 37.8%

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

      4. distribute-lft-in 37.8%

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

   7. Simplified 63.2%

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

      1. frac-2neg 63.2%

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

      2. sqrt-div 71.2%

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

   9. Applied egg-rr 71.2%

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

       1. frac-2neg 71.2%

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

       2. sqrt-div 84.5%

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

   11. Applied egg-rr 84.5%

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

   if -3.999999999999988e-310 < h

   1. Initial program 64.1%

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

   3. Simplified 63.4%

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

      1. associate-*r/ 65.5%

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

      2. *-commutative 65.5%

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

      3. div-inv 65.5%

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

      4. metadata-eval 65.5%

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

   6. Applied egg-rr 65.5%

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

   8. Applied egg-rr 75.8%

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

      1. unpow1 75.8%

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

      2. associate-*l/ 77.6%

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

      3. associate-/l* 76.0%

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

   10. Simplified 72.8%

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

       1. pow1/2 72.8%

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

       2. *-commutative 72.8%

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

       3. unpow-prod-down 83.5%

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

       4. pow1/2 83.5%

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

       5. pow1/2 83.5%

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

   12. Applied egg-rr 83.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M\_m \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_4 := 1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+133}:\\ \;\;\;\;t\_4 \cdot \left(t\_1 \cdot t\_2\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_3 \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+253}:\\ \;\;\;\;t\_1 \cdot \left(t\_2 \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(D \cdot \frac{M\_m}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot t\_4\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M_m D) (* d 2.0)) 2.0))))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fabs (/ d (sqrt (* h l)))))
        (t_4 (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l)))))
   (if (<= t_0 -5e+133)
     (* t_4 (* t_1 t_2))
     (if (<= t_0 0.0)
       (* t_3 (- 1.0 (* 0.5 (* (/ h l) (pow (* D (* M_m (/ 0.5 d))) 2.0)))))
       (if (<= t_0 2e+253)
         (*
          t_1
          (* t_2 (+ 1.0 (* h (* -0.125 (/ (pow (* D (/ M_m d)) 2.0) l))))))
         (* t_3 t_4))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M_m * D) / (d * 2.0)), 2.0))));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = fabs((d / sqrt((h * l))));
	double t_4 = 1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l));
	double tmp;
	if (t_0 <= -5e+133) {
		tmp = t_4 * (t_1 * t_2);
	} else if (t_0 <= 0.0) {
		tmp = t_3 * (1.0 - (0.5 * ((h / l) * pow((D * (M_m * (0.5 / d))), 2.0))));
	} else if (t_0 <= 2e+253) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (pow((D * (M_m / d)), 2.0) / l)))));
	} else {
		tmp = t_3 * t_4;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (((d / h) ** 0.5d0) * ((d / l) ** 0.5d0)) * (1.0d0 - ((h / l) * (0.5d0 * (((m_m * d_1) / (d * 2.0d0)) ** 2.0d0))))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    t_3 = abs((d / sqrt((h * l))))
    t_4 = 1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)) / l))
    if (t_0 <= (-5d+133)) then
        tmp = t_4 * (t_1 * t_2)
    else if (t_0 <= 0.0d0) then
        tmp = t_3 * (1.0d0 - (0.5d0 * ((h / l) * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0))))
    else if (t_0 <= 2d+253) then
        tmp = t_1 * (t_2 * (1.0d0 + (h * ((-0.125d0) * (((d_1 * (m_m / d)) ** 2.0d0) / l)))))
    else
        tmp = t_3 * t_4
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), 0.5) * Math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * Math.pow(((M_m * D) / (d * 2.0)), 2.0))));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double t_3 = Math.abs((d / Math.sqrt((h * l))));
	double t_4 = 1.0 - (0.5 * ((h * Math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l));
	double tmp;
	if (t_0 <= -5e+133) {
		tmp = t_4 * (t_1 * t_2);
	} else if (t_0 <= 0.0) {
		tmp = t_3 * (1.0 - (0.5 * ((h / l) * Math.pow((D * (M_m * (0.5 / d))), 2.0))));
	} else if (t_0 <= 2e+253) {
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (Math.pow((D * (M_m / d)), 2.0) / l)))));
	} else {
		tmp = t_3 * t_4;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), 0.5) * math.pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * math.pow(((M_m * D) / (d * 2.0)), 2.0))))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	t_3 = math.fabs((d / math.sqrt((h * l))))
	t_4 = 1.0 - (0.5 * ((h * math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l))
	tmp = 0
	if t_0 <= -5e+133:
		tmp = t_4 * (t_1 * t_2)
	elif t_0 <= 0.0:
		tmp = t_3 * (1.0 - (0.5 * ((h / l) * math.pow((D * (M_m * (0.5 / d))), 2.0))))
	elif t_0 <= 2e+253:
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (math.pow((D * (M_m / d)), 2.0) / l)))))
	else:
		tmp = t_3 * t_4
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M_m * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = abs(Float64(d / sqrt(Float64(h * l))))
	t_4 = Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l)))
	tmp = 0.0
	if (t_0 <= -5e+133)
		tmp = Float64(t_4 * Float64(t_1 * t_2));
	elseif (t_0 <= 0.0)
		tmp = Float64(t_3 * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)))));
	elseif (t_0 <= 2e+253)
		tmp = Float64(t_1 * Float64(t_2 * Float64(1.0 + Float64(h * Float64(-0.125 * Float64((Float64(D * Float64(M_m / d)) ^ 2.0) / l))))));
	else
		tmp = Float64(t_3 * t_4);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ 0.5) * ((d / l) ^ 0.5)) * (1.0 - ((h / l) * (0.5 * (((M_m * D) / (d * 2.0)) ^ 2.0))));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	t_3 = abs((d / sqrt((h * l))));
	t_4 = 1.0 - (0.5 * ((h * (((D / d) * (M_m * 0.5)) ^ 2.0)) / l));
	tmp = 0.0;
	if (t_0 <= -5e+133)
		tmp = t_4 * (t_1 * t_2);
	elseif (t_0 <= 0.0)
		tmp = t_3 * (1.0 - (0.5 * ((h / l) * ((D * (M_m * (0.5 / d))) ^ 2.0))));
	elseif (t_0 <= 2e+253)
		tmp = t_1 * (t_2 * (1.0 + (h * (-0.125 * (((D * (M_m / d)) ^ 2.0) / l)))));
	else
		tmp = t_3 * t_4;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+133], N[(t$95$4 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(t$95$3 * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+253], N[(t$95$1 * N[(t$95$2 * N[(1.0 + N[(h * N[(-0.125 * N[(N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * t$95$4), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999961e133

   1. Initial program 89.1%

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

   3. Simplified 86.7%

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

      1. associate-*r/ 88.6%

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

      2. *-commutative 88.6%

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

      3. div-inv 88.6%

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

      4. metadata-eval 88.6%

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

   6. Applied egg-rr 88.6%

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

   if -4.99999999999999961e133 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 54.2%

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

   3. Simplified 54.3%

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

      1. pow1 54.3%

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

      2. sqrt-unprod 44.2%

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

   6. Applied egg-rr 44.2%

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

      1. unpow1 44.2%

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

   8. Simplified 44.2%

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

      1. add-sqr-sqrt 44.2%

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

      2. rem-sqrt-square 44.2%

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

      3. frac-times 40.7%

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

      4. sqrt-div 44.7%

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

      5. sqrt-unprod 54.8%

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

      6. add-sqr-sqrt 89.2%

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

   10. Applied egg-rr 89.2%

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

   11. Taylor expanded in M around 0 89.1%

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

       1. associate-*r/ 89.1%

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

       2. *-commutative 89.1%

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

       3. associate-*l* 89.1%

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

       4. associate-*l/ 72.4%

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

       5. *-commutative 72.4%

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

       6. *-commutative 72.4%

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

       7. associate-/l* 72.4%

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

   13. Simplified 72.4%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.9999999999999999e253

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in h around -inf 69.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(-1 \cdot \left(h \cdot \left(0.125 \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} - \frac{1}{h}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 69.3%

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

      2. neg-mul-1 69.3%

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

      3. sub-neg 69.3%

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

      4. distribute-lft-in 69.3%

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

   7. Simplified 98.7%

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

      1. pow1 98.7%

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

      2. associate-*r/ 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. unpow1 98.7%

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

       2. distribute-lft-neg-in 98.7%

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

       3. distribute-rgt-neg-in 98.7%

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

       4. associate-/l* 98.7%

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

       5. distribute-lft-neg-in 98.7%

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

       6. metadata-eval 98.7%

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

       7. associate-*r/ 98.7%

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

       8. *-commutative 98.7%

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

       9. associate-/l* 98.7%

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

   11. Simplified 98.7%

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

   if 1.9999999999999999e253 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 14.9%

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

   3. Simplified 14.9%

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

      1. pow1 14.9%

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

      2. sqrt-unprod 13.8%

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

   6. Applied egg-rr 13.8%

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

      1. unpow1 13.8%

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

   8. Simplified 13.8%

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

      1. add-sqr-sqrt 13.8%

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

      2. rem-sqrt-square 13.8%

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

      3. frac-times 18.6%

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

      4. sqrt-div 23.2%

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

      5. sqrt-unprod 27.6%

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

      6. add-sqr-sqrt 47.3%

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

   10. Applied egg-rr 47.3%

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

       1. associate-*r/ 20.1%

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

       2. *-commutative 20.1%

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

       3. div-inv 20.1%

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

       4. metadata-eval 20.1%

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

   12. Applied egg-rr 60.5%

       \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2} \cdot h}{\ell}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq -5 \cdot 10^{+133}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 0:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + h \cdot \left(-0.125 \cdot \frac{{\left(D \cdot \frac{M}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M\_m}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M\_m \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -3.4e+106)
   (*
    (/ (sqrt (- d)) (sqrt (- l)))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* -0.125 (pow (* D (/ M_m d)) 2.0))))))
   (if (<= l 1.25e-142)
     (*
      (fabs (/ d (sqrt (* h l))))
      (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l))))
     (*
      d
      (/
       (fma (/ h l) (* -0.5 (pow (* D (/ (* M_m 0.5) d)) 2.0)) 1.0)
       (* (sqrt l) (sqrt h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -3.4e+106) {
		tmp = (sqrt(-d) / sqrt(-l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.125 * pow((D * (M_m / d)), 2.0)))));
	} else if (l <= 1.25e-142) {
		tmp = fabs((d / sqrt((h * l)))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else {
		tmp = d * (fma((h / l), (-0.5 * pow((D * ((M_m * 0.5) / d)), 2.0)), 1.0) / (sqrt(l) * sqrt(h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -3.4e+106)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.125 * (Float64(D * Float64(M_m / d)) ^ 2.0))))));
	elseif (l <= 1.25e-142)
		tmp = Float64(abs(Float64(d / sqrt(Float64(h * l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(Float64(M_m * 0.5) / d)) ^ 2.0)), 1.0) / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -3.4e+106], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.125 * N[Power[N[(D * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e-142], N[(N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.39999999999999994e106

   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%

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

   5. Taylor expanded in D around 0 48.7%

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

      1. *-commutative 48.7%

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

      2. associate-*l/ 48.7%

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

      3. associate-*r* 48.7%

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

   7. Simplified 48.7%

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

   8. Taylor expanded in D around 0 32.1%

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

      1. *-commutative 32.1%

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

      2. associate-/l* 34.5%

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

      3. unpow2 34.5%

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

      4. unpow2 34.5%

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

      5. unpow2 34.5%

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

      6. times-frac 39.5%

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

      7. swap-sqr 48.7%

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

      8. unpow2 48.7%

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

      9. associate-*r/ 48.7%

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

      10. *-commutative 48.7%

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

      11. associate-/l* 48.5%

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

   10. Simplified 48.5%

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

       1. frac-2neg 48.7%

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

       2. sqrt-div 61.4%

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

   12. Applied egg-rr 58.8%

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

   if -3.39999999999999994e106 < l < 1.2500000000000001e-142

   1. Initial program 69.1%

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

   3. Simplified 67.5%

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

      1. pow1 67.5%

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

      2. sqrt-unprod 56.1%

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

   6. Applied egg-rr 56.1%

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

      1. unpow1 56.1%

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

   8. Simplified 56.1%

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

      1. add-sqr-sqrt 56.1%

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

      2. rem-sqrt-square 56.1%

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

      3. frac-times 47.6%

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

      4. sqrt-div 52.2%

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

      5. sqrt-unprod 19.3%

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

      6. add-sqr-sqrt 77.1%

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

   10. Applied egg-rr 77.1%

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

       1. associate-*r/ 70.7%

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

       2. *-commutative 70.7%

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

       3. div-inv 70.7%

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

       4. metadata-eval 70.7%

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

   12. Applied egg-rr 85.1%

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

   if 1.2500000000000001e-142 < l

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

      1. associate-*r/ 63.4%

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

      2. *-commutative 63.4%

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

      3. div-inv 63.4%

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

      4. metadata-eval 63.4%

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

   6. Applied egg-rr 63.4%

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

   8. Applied egg-rr 74.7%

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

      1. unpow1 74.7%

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

      2. associate-*l/ 77.2%

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

      3. associate-/l* 77.1%

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

   10. Simplified 73.8%

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

       1. pow1/2 73.8%

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

       2. *-commutative 73.8%

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

       3. unpow-prod-down 84.3%

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

       4. pow1/2 84.3%

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

       5. pow1/2 84.3%

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

   12. Applied egg-rr 84.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.125 \cdot {\left(D \cdot \frac{M}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{-142}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+189}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.15e+148)
   (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
   (if (<= l 1.75e+189)
     (*
      (fabs (/ d (sqrt (* h l))))
      (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* M_m 0.5)) 2.0)) l))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.15e+148) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= 1.75e+189) {
		tmp = fabs((d / sqrt((h * l)))) * (1.0 - (0.5 * ((h * pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.15d+148)) then
        tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h))
    else if (l <= 1.75d+189) then
        tmp = abs((d / sqrt((h * l)))) * (1.0d0 - (0.5d0 * ((h * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)) / l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.15e+148) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h));
	} else if (l <= 1.75e+189) {
		tmp = Math.abs((d / Math.sqrt((h * l)))) * (1.0 - (0.5 * ((h * Math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1.15e+148:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))
	elif l <= 1.75e+189:
		tmp = math.fabs((d / math.sqrt((h * l)))) * (1.0 - (0.5 * ((h * math.pow(((D / d) * (M_m * 0.5)), 2.0)) / l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.15e+148)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= 1.75e+189)
		tmp = Float64(abs(Float64(d / sqrt(Float64(h * l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)) / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1.15e+148)
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	elseif (l <= 1.75e+189)
		tmp = abs((d / sqrt((h * l)))) * (1.0 - (0.5 * ((h * (((D / d) * (M_m * 0.5)) ^ 2.0)) / l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.15e+148], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+189], N[(N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.15e148

   1. Initial program 47.9%

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

   3. Simplified 47.7%

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

      1. frac-2neg 48.0%

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

      2. sqrt-div 60.9%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in d around inf 51.9%

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

   if -1.15e148 < l < 1.74999999999999998e189

   1. Initial program 67.5%

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

   3. Simplified 66.5%

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

      1. pow1 66.5%

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

      2. sqrt-unprod 53.8%

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

   6. Applied egg-rr 53.8%

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

      1. unpow1 53.8%

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

   8. Simplified 53.8%

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

      1. add-sqr-sqrt 53.8%

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

      2. rem-sqrt-square 53.8%

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

      3. frac-times 44.6%

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

      4. sqrt-div 49.1%

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

      5. sqrt-unprod 40.0%

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

      6. add-sqr-sqrt 77.3%

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

   10. Applied egg-rr 77.3%

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

       1. associate-*r/ 69.5%

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

       2. *-commutative 69.5%

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

       3. div-inv 69.5%

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

       4. metadata-eval 69.5%

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

   12. Applied egg-rr 83.3%

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

   if 1.74999999999999998e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+189}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{-145}:\\ \;\;\;\;\left|\frac{d}{t\_0}\right| \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M\_m \cdot 0.5}{d}\right)}^{2}, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -6.8e+137)
     (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
     (if (<= l 1.08e-145)
       (*
        (fabs (/ d t_0))
        (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0)))))
       (*
        d
        (/
         (fma (/ h l) (* -0.5 (pow (* D (/ (* M_m 0.5) d)) 2.0)) 1.0)
         t_0))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -6.8e+137) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= 1.08e-145) {
		tmp = fabs((d / t_0)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))));
	} else {
		tmp = d * (fma((h / l), (-0.5 * pow((D * ((M_m * 0.5) / d)), 2.0)), 1.0) / t_0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -6.8e+137)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= 1.08e-145)
		tmp = Float64(abs(Float64(d / t_0)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(Float64(M_m * 0.5) / d)) ^ 2.0)), 1.0) / t_0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -6.8e+137], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.08e-145], N[(N[Abs[N[(d / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.79999999999999973e137

   1. Initial program 47.9%

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

   3. Simplified 47.7%

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

      1. frac-2neg 48.0%

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

      2. sqrt-div 60.9%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in d around inf 51.9%

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

   if -6.79999999999999973e137 < l < 1.07999999999999998e-145

   1. Initial program 68.8%

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

   3. Simplified 67.3%

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

      1. pow1 67.3%

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

      2. sqrt-unprod 55.6%

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

   6. Applied egg-rr 55.6%

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

      1. unpow1 55.6%

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

   8. Simplified 55.6%

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

      1. add-sqr-sqrt 55.6%

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

      2. rem-sqrt-square 55.6%

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

      3. frac-times 47.3%

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

      4. sqrt-div 51.9%

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

      5. sqrt-unprod 18.9%

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

      6. add-sqr-sqrt 76.9%

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

   10. Applied egg-rr 76.9%

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

   if 1.07999999999999998e-145 < l

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

      1. associate-*r/ 63.4%

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

      2. *-commutative 63.4%

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

      3. div-inv 63.4%

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

      4. metadata-eval 63.4%

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

   6. Applied egg-rr 63.4%

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

   8. Applied egg-rr 74.7%

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

      1. unpow1 74.7%

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

      2. associate-*l/ 77.2%

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

      3. associate-/l* 77.1%

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

   10. Simplified 73.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+137}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{-145}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M\_m \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.45e+142)
   (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
   (if (<= l -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
     (*
      d
      (/
       (fma (/ h l) (* -0.5 (pow (* D (/ (* M_m 0.5) d)) 2.0)) 1.0)
       (sqrt (* h l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.45e+142) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else {
		tmp = d * (fma((h / l), (-0.5 * pow((D * ((M_m * 0.5) / d)), 2.0)), 1.0) / sqrt((h * l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.45e+142)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	else
		tmp = Float64(d * Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(Float64(M_m * 0.5) / d)) ^ 2.0)), 1.0) / sqrt(Float64(h * l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.45e+142], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(N[(M$95$m * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.45000000000000007e142

   1. Initial program 47.9%

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

   3. Simplified 47.7%

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

      1. frac-2neg 48.0%

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

      2. sqrt-div 60.9%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in d around inf 51.9%

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

   if -1.45000000000000007e142 < l < -4.999999999999985e-310

   1. Initial program 68.7%

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

   3. Simplified 67.7%

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

      1. pow1 67.7%

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

      2. sqrt-unprod 54.9%

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

   6. Applied egg-rr 54.9%

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

      1. unpow1 54.9%

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

   8. Simplified 54.9%

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

   9. Taylor expanded in d around -inf 77.3%

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

   if -4.999999999999985e-310 < l

   1. Initial program 64.1%

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

   3. Simplified 63.4%

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

      1. associate-*r/ 65.5%

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

      2. *-commutative 65.5%

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

      3. div-inv 65.5%

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

      4. metadata-eval 65.5%

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

   6. Applied egg-rr 65.5%

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

   8. Applied egg-rr 75.8%

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

      1. unpow1 75.8%

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

      2. associate-*l/ 77.6%

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

      3. associate-/l* 76.0%

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

   10. Simplified 72.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -7e+147)
   (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
   (*
    (fabs (/ d (sqrt (* h l))))
    (- 1.0 (* 0.5 (* (/ h l) (pow (* D (* M_m (/ 0.5 d))) 2.0)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -7e+147) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l)))) * (1.0 - (0.5 * ((h / l) * pow((D * (M_m * (0.5 / d))), 2.0))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-7d+147)) then
        tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h))
    else
        tmp = abs((d / sqrt((h * l)))) * (1.0d0 - (0.5d0 * ((h / l) * ((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -7e+147) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l)))) * (1.0 - (0.5 * ((h / l) * Math.pow((D * (M_m * (0.5 / d))), 2.0))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -7e+147:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / math.sqrt((h * l)))) * (1.0 - (0.5 * ((h / l) * math.pow((D * (M_m * (0.5 / d))), 2.0))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -7e+147)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(abs(Float64(d / sqrt(Float64(h * l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -7e+147)
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	else
		tmp = abs((d / sqrt((h * l)))) * (1.0 - (0.5 * ((h / l) * ((D * (M_m * (0.5 / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -7e+147], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.99999999999999949e147

   1. Initial program 47.9%

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

   3. Simplified 47.7%

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

      1. frac-2neg 48.0%

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

      2. sqrt-div 60.9%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in d around inf 51.9%

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

   if -6.99999999999999949e147 < l

   1. Initial program 66.2%

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

   3. Simplified 65.3%

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

      1. pow1 65.3%

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

      2. sqrt-unprod 52.4%

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

   6. Applied egg-rr 52.4%

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

      1. unpow1 52.4%

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

   8. Simplified 52.4%

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

      1. add-sqr-sqrt 52.4%

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

      2. rem-sqrt-square 52.4%

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

      3. frac-times 43.9%

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

      4. sqrt-div 48.9%

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

      5. sqrt-unprod 41.8%

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

      6. add-sqr-sqrt 76.0%

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

   10. Applied egg-rr 76.0%

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

   11. Taylor expanded in M around 0 76.9%

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

       1. associate-*r/ 76.9%

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

       2. *-commutative 76.9%

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

       3. associate-*l* 76.9%

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

       4. associate-*l/ 74.2%

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

       5. *-commutative 74.2%

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

       6. *-commutative 74.2%

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

       7. associate-/l* 74.2%

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

   13. Simplified 74.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right| \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -4.7e+147)
   (* (/ (sqrt (- d)) (sqrt (- l))) (sqrt (/ d h)))
   (if (<= l -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
     (if (<= l 1.8e+189)
       (*
        d
        (/
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0))))
         (sqrt (* h l))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.7e+147) {
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (l <= 1.8e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)))) / sqrt((h * l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-4.7d+147)) then
        tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h))
    else if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 1.8d+189) then
        tmp = d * ((1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)))) / sqrt((h * l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.7e+147) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-l)) * Math.sqrt((d / h));
	} else if (l <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (l <= 1.8e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -4.7e+147:
		tmp = (math.sqrt(-d) / math.sqrt(-l)) * math.sqrt((d / h))
	elif l <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0)
	elif l <= 1.8e+189:
		tmp = d * ((1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / math.sqrt((h * l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -4.7e+147)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * sqrt(Float64(d / h)));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	elseif (l <= 1.8e+189)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -4.7e+147)
		tmp = (sqrt(-d) / sqrt(-l)) * sqrt((d / h));
	elseif (l <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	elseif (l <= 1.8e+189)
		tmp = d * ((1.0 + ((h / l) * (-0.5 * (((D / d) * (M_m * 0.5)) ^ 2.0)))) / sqrt((h * l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -4.7e+147], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+189], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.7000000000000003e147

   1. Initial program 47.9%

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

   3. Simplified 47.7%

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

      1. frac-2neg 48.0%

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

      2. sqrt-div 60.9%

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

   6. Applied egg-rr 58.1%

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

   7. Taylor expanded in d around inf 51.9%

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

   if -4.7000000000000003e147 < l < -4.999999999999985e-310

   1. Initial program 68.7%

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

   3. Simplified 67.7%

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

      1. pow1 67.7%

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

      2. sqrt-unprod 54.9%

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

   6. Applied egg-rr 54.9%

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

      1. unpow1 54.9%

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

   8. Simplified 54.9%

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

   9. Taylor expanded in d around -inf 77.3%

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

   if -4.999999999999985e-310 < l < 1.80000000000000004e189

   1. Initial program 66.4%

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

   3. Simplified 65.5%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. div-inv 68.9%

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

      4. metadata-eval 68.9%

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

   6. Applied egg-rr 68.9%

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

   8. Applied egg-rr 78.1%

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

      1. unpow1 78.1%

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

      2. associate-*l/ 80.3%

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

      3. associate-/l* 78.4%

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

   10. Simplified 75.6%

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

       1. fma-undefine 75.6%

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

       2. associate-*r/ 78.4%

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

       3. associate-*l/ 77.5%

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

       4. *-commutative 77.5%

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

   12. Applied egg-rr 77.5%

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

   if 1.80000000000000004e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -6.4e+147)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -5e-310)
     (*
      (* d (sqrt (/ 1.0 (* h l))))
      (+ (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))) -1.0))
     (if (<= l 2.75e+189)
       (*
        d
        (/
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0))))
         (sqrt (* h l))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -6.4e+147) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= -5e-310) {
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (l <= 2.75e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)))) / sqrt((h * l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6.4d+147)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= (-5d-310)) then
        tmp = (d * sqrt((1.0d0 / (h * l)))) * ((0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0))) + (-1.0d0))
    else if (l <= 2.75d+189) then
        tmp = d * ((1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)))) / sqrt((h * l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -6.4e+147) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= -5e-310) {
		tmp = (d * Math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0);
	} else if (l <= 2.75e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -6.4e+147:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= -5e-310:
		tmp = (d * math.sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0))) + -1.0)
	elif l <= 2.75e+189:
		tmp = d * ((1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / math.sqrt((h * l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -6.4e+147)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= -5e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(h * l)))) * Float64(Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))) + -1.0));
	elseif (l <= 2.75e+189)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -6.4e+147)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= -5e-310)
		tmp = (d * sqrt((1.0 / (h * l)))) * ((0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0))) + -1.0);
	elseif (l <= 2.75e+189)
		tmp = d * ((1.0 + ((h / l) * (-0.5 * (((D / d) * (M_m * 0.5)) ^ 2.0)))) / sqrt((h * l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -6.4e+147], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.75e+189], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.39999999999999958e147

   1. Initial program 47.9%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in d around inf 5.4%

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

      1. pow1 5.4%

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

      2. pow1/2 5.4%

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

      3. inv-pow 5.4%

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

      4. pow-pow 5.4%

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

      5. metadata-eval 5.4%

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

   7. Applied egg-rr 5.4%

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

      1. unpow1 5.4%

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

   9. Simplified 5.4%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 48.4%

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

       6. neg-mul-1 48.4%

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

   12. Simplified 48.4%

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

   if -6.39999999999999958e147 < l < -4.999999999999985e-310

   1. Initial program 68.7%

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

   3. Simplified 67.7%

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

      1. pow1 67.7%

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

      2. sqrt-unprod 54.9%

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

   6. Applied egg-rr 54.9%

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

      1. unpow1 54.9%

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

   8. Simplified 54.9%

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

   9. Taylor expanded in d around -inf 77.3%

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

   if -4.999999999999985e-310 < l < 2.75e189

   1. Initial program 66.4%

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

   3. Simplified 65.5%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. div-inv 68.9%

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

      4. metadata-eval 68.9%

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

   6. Applied egg-rr 68.9%

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

   8. Applied egg-rr 78.1%

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

      1. unpow1 78.1%

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

      2. associate-*l/ 80.3%

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

      3. associate-/l* 78.4%

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

   10. Simplified 75.6%

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

       1. fma-undefine 75.6%

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

       2. associate-*r/ 78.4%

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

       3. associate-*l/ 77.5%

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

       4. *-commutative 77.5%

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

   12. Applied egg-rr 77.5%

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

   if 2.75e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+147}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot \left(0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-259}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot t\_0}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot t\_0\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* (/ D d) (* M_m 0.5)) 2.0)))
   (if (<= l -1.02e+86)
     (* d (- (sqrt (/ (/ 1.0 h) l))))
     (if (<= l 7.5e-259)
       (* (- 1.0 (* 0.5 (/ (* h t_0) l))) (sqrt (* (/ d h) (/ d l))))
       (if (<= l 1.75e+189)
         (* d (/ (+ 1.0 (* (/ h l) (* -0.5 t_0))) (sqrt (* h l))))
         (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow(((D / d) * (M_m * 0.5)), 2.0);
	double tmp;
	if (l <= -1.02e+86) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 7.5e-259) {
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * sqrt(((d / h) * (d / l)));
	} else if (l <= 1.75e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * t_0))) / sqrt((h * l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0
    if (l <= (-1.02d+86)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 7.5d-259) then
        tmp = (1.0d0 - (0.5d0 * ((h * t_0) / l))) * sqrt(((d / h) * (d / l)))
    else if (l <= 1.75d+189) then
        tmp = d * ((1.0d0 + ((h / l) * ((-0.5d0) * t_0))) / sqrt((h * l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow(((D / d) * (M_m * 0.5)), 2.0);
	double tmp;
	if (l <= -1.02e+86) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 7.5e-259) {
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 1.75e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * t_0))) / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow(((D / d) * (M_m * 0.5)), 2.0)
	tmp = 0
	if l <= -1.02e+86:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 7.5e-259:
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * math.sqrt(((d / h) * (d / l)))
	elif l <= 1.75e+189:
		tmp = d * ((1.0 + ((h / l) * (-0.5 * t_0))) / math.sqrt((h * l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0
	tmp = 0.0
	if (l <= -1.02e+86)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 7.5e-259)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h * t_0) / l))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 1.75e+189)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * t_0))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = ((D / d) * (M_m * 0.5)) ^ 2.0;
	tmp = 0.0;
	if (l <= -1.02e+86)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 7.5e-259)
		tmp = (1.0 - (0.5 * ((h * t_0) / l))) * sqrt(((d / h) * (d / l)));
	elseif (l <= 1.75e+189)
		tmp = d * ((1.0 + ((h / l) * (-0.5 * t_0))) / sqrt((h * l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[l, -1.02e+86], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 7.5e-259], N[(N[(1.0 - N[(0.5 * N[(N[(h * t$95$0), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.75e+189], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.01999999999999996e86

   1. Initial program 45.7%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in d around inf 4.7%

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

      1. pow1 4.7%

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

      2. pow1/2 4.7%

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

      3. inv-pow 4.7%

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

      4. pow-pow 4.7%

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

      5. metadata-eval 4.7%

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

   7. Applied egg-rr 4.7%

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

      1. unpow1 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 46.8%

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

       6. neg-mul-1 46.8%

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

   12. Simplified 46.8%

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

   if -1.01999999999999996e86 < l < 7.50000000000000052e-259

   1. Initial program 72.4%

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

   3. Simplified 71.4%

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

      1. associate-*r/ 73.9%

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

      2. *-commutative 73.9%

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

      3. div-inv 73.9%

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

      4. metadata-eval 73.9%

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

   6. Applied egg-rr 73.9%

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

      1. pow1 71.4%

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

      2. sqrt-unprod 59.2%

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

   8. Applied egg-rr 61.7%

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

      1. unpow1 59.2%

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

   10. Simplified 61.7%

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

   if 7.50000000000000052e-259 < l < 1.74999999999999998e189

   1. Initial program 65.4%

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

   3. Simplified 64.4%

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

      1. associate-*r/ 68.0%

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

      2. *-commutative 68.0%

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

      3. div-inv 68.0%

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

      4. metadata-eval 68.0%

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

   6. Applied egg-rr 68.0%

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

   8. Applied egg-rr 80.7%

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

      1. unpow1 80.7%

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

      2. associate-*l/ 83.0%

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

      3. associate-/l* 81.0%

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

   10. Simplified 78.0%

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

       1. fma-undefine 78.0%

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

       2. associate-*r/ 81.0%

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

       3. associate-*l/ 80.0%

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

       4. *-commutative 80.0%

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

   12. Applied egg-rr 80.0%

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

   if 1.74999999999999998e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-259}:\\ \;\;\;\;\left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}}{\ell}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+88}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M\_m \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -4.6e+88)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 1.45e-240)
     (*
      (sqrt (* (/ d h) (/ d l)))
      (- 1.0 (* 0.5 (* (/ h l) (pow (/ (* M_m 0.5) (/ d D)) 2.0)))))
     (if (<= l 1.8e+189)
       (*
        d
        (/
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0))))
         (sqrt (* h l))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.6e+88) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 1.45e-240) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * pow(((M_m * 0.5) / (d / D)), 2.0))));
	} else if (l <= 1.8e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)))) / sqrt((h * l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-4.6d+88)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 1.45d-240) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 - (0.5d0 * ((h / l) * (((m_m * 0.5d0) / (d / d_1)) ** 2.0d0))))
    else if (l <= 1.8d+189) then
        tmp = d * ((1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)))) / sqrt((h * l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.6e+88) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 1.45e-240) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * Math.pow(((M_m * 0.5) / (d / D)), 2.0))));
	} else if (l <= 1.8e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -4.6e+88:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 1.45e-240:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * math.pow(((M_m * 0.5) / (d / D)), 2.0))))
	elif l <= 1.8e+189:
		tmp = d * ((1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / math.sqrt((h * l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -4.6e+88)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 1.45e-240)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M_m * 0.5) / Float64(d / D)) ^ 2.0)))));
	elseif (l <= 1.8e+189)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -4.6e+88)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 1.45e-240)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 - (0.5 * ((h / l) * (((M_m * 0.5) / (d / D)) ^ 2.0))));
	elseif (l <= 1.8e+189)
		tmp = d * ((1.0 + ((h / l) * (-0.5 * (((D / d) * (M_m * 0.5)) ^ 2.0)))) / sqrt((h * l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -4.6e+88], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 1.45e-240], N[(N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M$95$m * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+189], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.6000000000000003e88

   1. Initial program 45.7%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in d around inf 4.7%

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

      1. pow1 4.7%

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

      2. pow1/2 4.7%

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

      3. inv-pow 4.7%

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

      4. pow-pow 4.7%

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

      5. metadata-eval 4.7%

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

   7. Applied egg-rr 4.7%

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

      1. unpow1 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 46.8%

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

       6. neg-mul-1 46.8%

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

   12. Simplified 46.8%

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

   if -4.6000000000000003e88 < l < 1.4500000000000001e-240

   1. Initial program 73.3%

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

   3. Simplified 72.3%

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

      1. pow1 72.3%

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

      2. sqrt-unprod 59.5%

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

   6. Applied egg-rr 59.5%

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

      1. unpow1 59.5%

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

   8. Simplified 59.5%

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

      1. clear-num 59.5%

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

      2. un-div-inv 59.5%

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

      3. div-inv 59.5%

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

      4. metadata-eval 59.5%

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

      5. *-commutative 59.5%

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

   10. Applied egg-rr 59.5%

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

   if 1.4500000000000001e-240 < l < 1.80000000000000004e189

   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 63.3%

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

      1. associate-*r/ 67.0%

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

      2. *-commutative 67.0%

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

      3. div-inv 67.0%

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

      4. metadata-eval 67.0%

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

   6. Applied egg-rr 67.0%

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

   8. Applied egg-rr 81.1%

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

      1. unpow1 81.1%

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

      2. associate-*l/ 83.4%

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

      3. associate-/l* 81.4%

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

   10. Simplified 78.3%

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

       1. fma-undefine 78.3%

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

       2. associate-*r/ 81.4%

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

       3. associate-*l/ 80.4%

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

       4. *-commutative 80.4%

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

   12. Applied egg-rr 80.4%

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

   if 1.80000000000000004e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+88}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+85}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-262}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.8e+85)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 9.5e-262)
     (*
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M_m 2.0)) 2.0))))
      (sqrt (* (/ d h) (/ d l))))
     (if (<= l 2.75e+189)
       (*
        d
        (/
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0))))
         (sqrt (* h l))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.8e+85) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 9.5e-262) {
		tmp = (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M_m / 2.0)), 2.0)))) * sqrt(((d / h) * (d / l)));
	} else if (l <= 2.75e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)))) / sqrt((h * l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.8d+85)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 9.5d-262) then
        tmp = (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0)))) * sqrt(((d / h) * (d / l)))
    else if (l <= 2.75d+189) then
        tmp = d * ((1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m_m * 0.5d0)) ** 2.0d0)))) / sqrt((h * l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.8e+85) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 9.5e-262) {
		tmp = (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M_m / 2.0)), 2.0)))) * Math.sqrt(((d / h) * (d / l)));
	} else if (l <= 2.75e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / Math.sqrt((h * l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1.8e+85:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 9.5e-262:
		tmp = (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M_m / 2.0)), 2.0)))) * math.sqrt(((d / h) * (d / l)))
	elif l <= 2.75e+189:
		tmp = d * ((1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M_m * 0.5)), 2.0)))) / math.sqrt((h * l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.8e+85)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 9.5e-262)
		tmp = Float64(Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0)))) * sqrt(Float64(Float64(d / h) * Float64(d / l))));
	elseif (l <= 2.75e+189)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1.8e+85)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 9.5e-262)
		tmp = (1.0 - (0.5 * ((h / l) * (((D / d) * (M_m / 2.0)) ^ 2.0)))) * sqrt(((d / h) * (d / l)));
	elseif (l <= 2.75e+189)
		tmp = d * ((1.0 + ((h / l) * (-0.5 * (((D / d) * (M_m * 0.5)) ^ 2.0)))) / sqrt((h * l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.8e+85], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 9.5e-262], N[(N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.75e+189], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.7999999999999999e85

   1. Initial program 45.7%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in d around inf 4.7%

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

      1. pow1 4.7%

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

      2. pow1/2 4.7%

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

      3. inv-pow 4.7%

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

      4. pow-pow 4.7%

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

      5. metadata-eval 4.7%

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

   7. Applied egg-rr 4.7%

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

      1. unpow1 4.7%

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

   9. Simplified 4.7%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 46.8%

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

       6. neg-mul-1 46.8%

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

   12. Simplified 46.8%

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

   if -1.7999999999999999e85 < l < 9.4999999999999999e-262

   1. Initial program 72.4%

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

   3. Simplified 71.4%

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

      1. pow1 71.4%

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

      2. sqrt-unprod 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. unpow1 59.2%

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

   8. Simplified 59.2%

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

   if 9.4999999999999999e-262 < l < 2.75e189

   1. Initial program 65.4%

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

   3. Simplified 64.4%

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

      1. associate-*r/ 68.0%

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

      2. *-commutative 68.0%

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

      3. div-inv 68.0%

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

      4. metadata-eval 68.0%

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

   6. Applied egg-rr 68.0%

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

   8. Applied egg-rr 80.7%

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

      1. unpow1 80.7%

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

      2. associate-*l/ 83.0%

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

      3. associate-/l* 81.0%

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

   10. Simplified 78.0%

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

       1. fma-undefine 78.0%

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

       2. associate-*r/ 81.0%

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

       3. associate-*l/ 80.0%

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

       4. *-commutative 80.0%

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

   12. Applied egg-rr 80.0%

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

   if 2.75e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{+85}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-262}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-69}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M\_m \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.9e-69)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l -5e-310)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (if (<= l 1.95e+189)
       (*
        d
        (/
         (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (* M_m 0.5)) 2.0))))
         (sqrt (* h l))))
       (* d (/ (sqrt (/ 1.0 h)) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.9e-69) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= -5e-310) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else if (l <= 1.95e+189) {
		tmp = d * ((1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m * 0.5)), 2.0)))) / sqrt((h * l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.9e-69)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= -5e-310)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	elseif (l <= 1.95e+189)
		tmp = Float64(d * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m * 0.5)) ^ 2.0)))) / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.9e-69], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e+189], N[(d * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.8999999999999999e-69

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in d around inf 4.1%

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

      1. pow1 4.1%

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

      2. pow1/2 4.1%

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

      3. inv-pow 4.1%

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

      4. pow-pow 4.1%

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

      5. metadata-eval 4.1%

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

   7. Applied egg-rr 4.1%

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

      1. unpow1 4.1%

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

   9. Simplified 4.1%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 40.7%

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

       6. neg-mul-1 40.7%

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

   12. Simplified 40.7%

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

   if -1.8999999999999999e-69 < l < -4.999999999999985e-310

   1. Initial program 72.8%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in d around inf 24.4%

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

      1. pow1 24.4%

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

      2. pow1/2 24.4%

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

      3. inv-pow 24.4%

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

      4. pow-pow 22.7%

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

      5. metadata-eval 22.7%

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

   7. Applied egg-rr 22.7%

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

      1. unpow1 22.7%

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

   9. Simplified 22.7%

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

       1. expm1-log1p-u 22.7%

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

       2. expm1-undefine 59.4%

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

   11. Applied egg-rr 59.4%

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

       1. sub-neg 59.4%

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

       2. metadata-eval 59.4%

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

       3. +-commutative 59.4%

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

       4. log1p-undefine 59.4%

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

       5. rem-exp-log 59.4%

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

       6. +-commutative 59.4%

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

       7. fma-define 59.4%

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

   13. Simplified 59.4%

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

   if -4.999999999999985e-310 < l < 1.95e189

   1. Initial program 66.4%

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

   3. Simplified 65.5%

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

      1. associate-*r/ 68.9%

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

      2. *-commutative 68.9%

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

      3. div-inv 68.9%

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

      4. metadata-eval 68.9%

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

   6. Applied egg-rr 68.9%

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

   8. Applied egg-rr 78.1%

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

      1. unpow1 78.1%

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

      2. associate-*l/ 80.3%

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

      3. associate-/l* 78.4%

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

   10. Simplified 75.6%

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

       1. fma-undefine 75.6%

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

       2. associate-*r/ 78.4%

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

       3. associate-*l/ 77.5%

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

       4. *-commutative 77.5%

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

   12. Applied egg-rr 77.5%

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

   if 1.95e189 < l

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in d around inf 67.7%

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

      1. pow1 67.7%

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

      2. pow1/2 67.7%

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

      3. inv-pow 67.7%

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

      4. pow-pow 67.8%

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

      5. metadata-eval 67.8%

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

   7. Applied egg-rr 67.8%

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

      1. unpow1 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in d around 0 67.7%

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

       1. associate-/r* 67.9%

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

   12. Simplified 67.9%

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

       1. sqrt-div 83.6%

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

   14. Applied egg-rr 83.6%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-69}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+189}:\\ \;\;\;\;d \cdot \frac{1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-73}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.8e-73)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 1.65e-280)
     (* d (pow (+ -1.0 (fma h l 1.0)) -0.5))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.8e-73) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 1.65e-280) {
		tmp = d * pow((-1.0 + fma(h, l, 1.0)), -0.5);
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.8e-73)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 1.65e-280)
		tmp = Float64(d * (Float64(-1.0 + fma(h, l, 1.0)) ^ -0.5));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.8e-73], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 1.65e-280], N[(d * N[Power[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.8e-73

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in d around inf 4.1%

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

      1. pow1 4.1%

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

      2. pow1/2 4.1%

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

      3. inv-pow 4.1%

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

      4. pow-pow 4.1%

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

      5. metadata-eval 4.1%

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

   7. Applied egg-rr 4.1%

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

      1. unpow1 4.1%

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

   9. Simplified 4.1%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 40.7%

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

       6. neg-mul-1 40.7%

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

   12. Simplified 40.7%

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

   if -1.8e-73 < l < 1.64999999999999995e-280

   1. Initial program 72.5%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in d around inf 24.9%

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

      1. pow1 24.9%

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

      2. pow1/2 24.9%

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

      3. inv-pow 24.9%

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

      4. pow-pow 23.2%

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

      5. metadata-eval 23.2%

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

   7. Applied egg-rr 23.2%

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

      1. unpow1 23.2%

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

   9. Simplified 23.2%

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

       1. expm1-log1p-u 23.2%

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

       2. expm1-undefine 58.0%

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

   11. Applied egg-rr 58.0%

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

       1. sub-neg 58.0%

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

       2. metadata-eval 58.0%

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

       3. +-commutative 58.0%

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

       4. log1p-undefine 58.0%

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

       5. rem-exp-log 58.1%

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

       6. +-commutative 58.1%

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

       7. fma-define 58.1%

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

   13. Simplified 58.1%

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

   if 1.64999999999999995e-280 < l

   1. Initial program 64.1%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in d around inf 45.6%

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

      1. pow1 45.6%

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

      2. pow1/2 45.6%

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

      3. inv-pow 45.6%

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

      4. pow-pow 45.6%

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

      5. metadata-eval 45.6%

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

   7. Applied egg-rr 45.6%

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

      1. unpow1 45.6%

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

   9. Simplified 45.6%

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

   10. Taylor expanded in d around 0 45.6%

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

       1. associate-/r* 45.6%

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

   12. Simplified 45.6%

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

       1. sqrt-div 54.3%

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

   14. Applied egg-rr 54.3%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8 \cdot 10^{-73}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;d \cdot {\left(-1 + \mathsf{fma}\left(h, \ell, 1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-183}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-283}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -1.05e-183)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 7.2e-283)
     (- (sqrt (* (/ d h) (/ d l))))
     (* d (/ (sqrt (/ 1.0 h)) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.05e-183) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 7.2e-283) {
		tmp = -sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.05d-183)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 7.2d-283) then
        tmp = -sqrt(((d / h) * (d / l)))
    else
        tmp = d * (sqrt((1.0d0 / h)) / sqrt(l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -1.05e-183) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 7.2e-283) {
		tmp = -Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (Math.sqrt((1.0 / h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -1.05e-183:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 7.2e-283:
		tmp = -math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * (math.sqrt((1.0 / h)) / math.sqrt(l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -1.05e-183)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 7.2e-283)
		tmp = Float64(-sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * Float64(sqrt(Float64(1.0 / h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -1.05e-183)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 7.2e-283)
		tmp = -sqrt(((d / h) * (d / l)));
	else
		tmp = d * (sqrt((1.0 / h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -1.05e-183], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 7.2e-283], (-N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(d * N[(N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.0500000000000001e-183

   1. Initial program 59.9%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in d around inf 5.5%

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

      1. pow1 5.5%

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

      2. pow1/2 5.5%

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

      3. inv-pow 5.5%

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

      4. pow-pow 4.6%

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

      5. metadata-eval 4.6%

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

   7. Applied egg-rr 4.6%

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

      1. unpow1 4.6%

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

   9. Simplified 4.6%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 39.5%

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

       6. neg-mul-1 39.5%

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

   12. Simplified 39.5%

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

   if -1.0500000000000001e-183 < l < 7.1999999999999999e-283

   1. Initial program 73.6%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 37.3%

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

      4. mul-1-neg 37.3%

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

   7. Simplified 37.3%

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

      1. pow1 37.3%

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

      2. distribute-rgt-neg-out 37.3%

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

      3. sqrt-unprod 45.6%

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

   9. Applied egg-rr 45.6%

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

       1. unpow1 45.6%

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

       2. *-commutative 45.6%

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

   11. Simplified 45.6%

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

   if 7.1999999999999999e-283 < l

   1. Initial program 63.5%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in d around inf 46.1%

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

      1. pow1 46.1%

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

      2. pow1/2 46.1%

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

      3. inv-pow 46.1%

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

      4. pow-pow 46.1%

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

      5. metadata-eval 46.1%

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

   7. Applied egg-rr 46.1%

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

      1. unpow1 46.1%

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

   9. Simplified 46.1%

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

   10. Taylor expanded in d around 0 46.1%

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

       1. associate-/r* 46.1%

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

   12. Simplified 46.1%

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

       1. sqrt-div 54.6%

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

   14. Applied egg-rr 54.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{-183}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-283}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\sqrt{\frac{1}{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-283}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -6e-184)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 7.2e-283)
     (- (sqrt (* (/ d h) (/ d l))))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -6e-184) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 7.2e-283) {
		tmp = -sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6d-184)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 7.2d-283) then
        tmp = -sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((l ** (-0.5d0)) * (h ** (-0.5d0)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -6e-184) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 7.2e-283) {
		tmp = -Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -6e-184:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 7.2e-283:
		tmp = -math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * (math.pow(l, -0.5) * math.pow(h, -0.5))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -6e-184)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 7.2e-283)
		tmp = Float64(-sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -6e-184)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 7.2e-283)
		tmp = -sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((l ^ -0.5) * (h ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -6e-184], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 7.2e-283], (-N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.99999999999999982e-184

   1. Initial program 59.9%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in d around inf 5.5%

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

      1. pow1 5.5%

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

      2. pow1/2 5.5%

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

      3. inv-pow 5.5%

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

      4. pow-pow 4.6%

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

      5. metadata-eval 4.6%

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

   7. Applied egg-rr 4.6%

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

      1. unpow1 4.6%

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

   9. Simplified 4.6%

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

   10. Taylor expanded in h around -inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/r* 0.0%

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

       3. *-commutative 0.0%

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

       4. unpow2 0.0%

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

       5. rem-square-sqrt 39.5%

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

       6. neg-mul-1 39.5%

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

   12. Simplified 39.5%

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

   if -5.99999999999999982e-184 < l < 7.1999999999999999e-283

   1. Initial program 73.6%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 37.3%

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

      4. mul-1-neg 37.3%

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

   7. Simplified 37.3%

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

      1. pow1 37.3%

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

      2. distribute-rgt-neg-out 37.3%

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

      3. sqrt-unprod 45.6%

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

   9. Applied egg-rr 45.6%

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

       1. unpow1 45.6%

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

       2. *-commutative 45.6%

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

   11. Simplified 45.6%

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

   if 7.1999999999999999e-283 < l

   1. Initial program 63.5%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in d around inf 46.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1 46.1%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 46.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 46.1%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. pow-pow 46.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      5. metadata-eval 46.1%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   7. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 46.1%

         \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   10. Step-by-step derivation

       1. *-commutative 46.1%

          \[\leadsto d \cdot {\color{blue}{\left(\ell \cdot h\right)}}^{-0.5} \]

       2. unpow-prod-down 54.6%

          \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]

   11. Applied egg-rr 54.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-283}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-281}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= l -4.1e-184)
   (* d (- (sqrt (/ (/ 1.0 h) l))))
   (if (<= l 1.3e-281)
     (- (sqrt (* (/ d h) (/ d l))))
     (* d (pow (* h l) -0.5)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.1e-184) {
		tmp = d * -sqrt(((1.0 / h) / l));
	} else if (l <= 1.3e-281) {
		tmp = -sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * pow((h * l), -0.5);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-4.1d-184)) then
        tmp = d * -sqrt(((1.0d0 / h) / l))
    else if (l <= 1.3d-281) then
        tmp = -sqrt(((d / h) * (d / l)))
    else
        tmp = d * ((h * l) ** (-0.5d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (l <= -4.1e-184) {
		tmp = d * -Math.sqrt(((1.0 / h) / l));
	} else if (l <= 1.3e-281) {
		tmp = -Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * Math.pow((h * l), -0.5);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if l <= -4.1e-184:
		tmp = d * -math.sqrt(((1.0 / h) / l))
	elif l <= 1.3e-281:
		tmp = -math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * math.pow((h * l), -0.5)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (l <= -4.1e-184)
		tmp = Float64(d * Float64(-sqrt(Float64(Float64(1.0 / h) / l))));
	elseif (l <= 1.3e-281)
		tmp = Float64(-sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * (Float64(h * l) ^ -0.5));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (l <= -4.1e-184)
		tmp = d * -sqrt(((1.0 / h) / l));
	elseif (l <= 1.3e-281)
		tmp = -sqrt(((d / h) * (d / l)));
	else
		tmp = d * ((h * l) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[l, -4.1e-184], N[(d * (-N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[l, 1.3e-281], (-N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.1e-184

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 5.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1 5.5%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 5.5%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 5.5%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. pow-pow 4.6%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      5. metadata-eval 4.6%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   7. Applied egg-rr 4.6%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 4.6%

         \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 4.6%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Taylor expanded in h around -inf 0.0%

       \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]

       2. associate-/r* 0.0%

          \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \]

       3. *-commutative 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \]

       4. unpow2 0.0%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \]

       5. rem-square-sqrt 39.5%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(\color{blue}{-1} \cdot d\right) \]

       6. neg-mul-1 39.5%

          \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \color{blue}{\left(-d\right)} \]

   12. Simplified 39.5%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)} \]

   if -4.1e-184 < l < 1.30000000000000002e-281

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. rem-square-sqrt 37.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. mul-1-neg 37.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

   7. Simplified 37.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]
   8. Step-by-step derivation

      1. pow1 37.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(-\sqrt{\frac{d}{\ell}}\right)\right)}^{1}} \]

      2. distribute-rgt-neg-out 37.3%

         \[\leadsto {\color{blue}{\left(-\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}}^{1} \]

      3. sqrt-unprod 45.6%

         \[\leadsto {\left(-\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

   9. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{{\left(-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 45.6%

          \[\leadsto \color{blue}{-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       2. *-commutative 45.6%

          \[\leadsto -\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   11. Simplified 45.6%

       \[\leadsto \color{blue}{-\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if 1.30000000000000002e-281 < l

   1. Initial program 63.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 46.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1 46.1%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 46.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 46.1%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. pow-pow 46.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      5. metadata-eval 46.1%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   7. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 46.1%

         \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-\sqrt{\frac{\frac{1}{h}}{\ell}}\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-281}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-282}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -2.9e-184)
     (* d (- t_0))
     (if (<= l 1.45e-282) (- (sqrt (* (/ d h) (/ d l)))) (* d t_0)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -2.9e-184) {
		tmp = d * -t_0;
	} else if (l <= 1.45e-282) {
		tmp = -sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-2.9d-184)) then
        tmp = d * -t_0
    else if (l <= 1.45d-282) then
        tmp = -sqrt(((d / h) * (d / l)))
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -2.9e-184) {
		tmp = d * -t_0;
	} else if (l <= 1.45e-282) {
		tmp = -Math.sqrt(((d / h) * (d / l)));
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -2.9e-184:
		tmp = d * -t_0
	elif l <= 1.45e-282:
		tmp = -math.sqrt(((d / h) * (d / l)))
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -2.9e-184)
		tmp = Float64(d * Float64(-t_0));
	elseif (l <= 1.45e-282)
		tmp = Float64(-sqrt(Float64(Float64(d / h) * Float64(d / l))));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -2.9e-184)
		tmp = d * -t_0;
	elseif (l <= 1.45e-282)
		tmp = -sqrt(((d / h) * (d / l)));
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -2.9e-184], N[(d * (-t$95$0)), $MachinePrecision], If[LessEqual[l, 1.45e-282], (-N[Sqrt[N[(N[(d / h), $MachinePrecision] * N[(d / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(d * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.90000000000000014e-184

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 5.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d} \]

      3. unpow2 0.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      4. rem-square-sqrt 38.1%

         \[\leadsto \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. *-commutative 38.1%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)} \cdot d \]

      6. associate-*r* 38.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      7. unpow1/2 38.1%

         \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot \left(-1 \cdot d\right) \]

      8. rem-exp-log 36.2%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(-1 \cdot d\right) \]

      9. exp-neg 36.2%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(-1 \cdot d\right) \]

      10. exp-prod 36.2%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      11. distribute-lft-neg-out 36.2%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      12. distribute-rgt-neg-in 36.2%

          \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(-1 \cdot d\right) \]

      13. metadata-eval 36.2%

          \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(-1 \cdot d\right) \]

      14. exp-to-pow 38.1%

          \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-1 \cdot d\right) \]

      15. neg-mul-1 38.1%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 38.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -2.90000000000000014e-184 < l < 1.44999999999999999e-282

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in l around -inf 0.0%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      2. unpow2 0.0%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      3. rem-square-sqrt 37.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      4. mul-1-neg 37.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

   7. Simplified 37.3%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]
   8. Step-by-step derivation

      1. pow1 37.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h}} \cdot \left(-\sqrt{\frac{d}{\ell}}\right)\right)}^{1}} \]

      2. distribute-rgt-neg-out 37.3%

         \[\leadsto {\color{blue}{\left(-\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)}}^{1} \]

      3. sqrt-unprod 45.6%

         \[\leadsto {\left(-\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}}\right)}^{1} \]

   9. Applied egg-rr 45.6%

      \[\leadsto \color{blue}{{\left(-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 45.6%

          \[\leadsto \color{blue}{-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       2. *-commutative 45.6%

          \[\leadsto -\sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   11. Simplified 45.6%

       \[\leadsto \color{blue}{-\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   if 1.44999999999999999e-282 < l

   1. Initial program 63.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 46.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1 46.1%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 46.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 46.1%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. pow-pow 46.1%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      5. metadata-eval 46.1%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   7. Applied egg-rr 46.1%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 46.1%

         \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-282}:\\ \;\;\;\;-\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(h \cdot \ell\right)}^{-0.5}\\ \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (pow (* h l) -0.5)))
   (if (<= l -1.6e-184) (* d (- t_0)) (* d t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((h * l), -0.5);
	double tmp;
	if (l <= -1.6e-184) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (h * l) ** (-0.5d0)
    if (l <= (-1.6d-184)) then
        tmp = d * -t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = Math.pow((h * l), -0.5);
	double tmp;
	if (l <= -1.6e-184) {
		tmp = d * -t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = math.pow((h * l), -0.5)
	tmp = 0
	if l <= -1.6e-184:
		tmp = d * -t_0
	else:
		tmp = d * t_0
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(h * l) ^ -0.5
	tmp = 0.0
	if (l <= -1.6e-184)
		tmp = Float64(d * Float64(-t_0));
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (h * l) ^ -0.5;
	tmp = 0.0;
	if (l <= -1.6e-184)
		tmp = d * -t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[l, -1.6e-184], N[(d * (-t$95$0)), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.6e-184

   1. Initial program 59.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 5.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   6. Taylor expanded in h around -inf 0.0%

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   7. Step-by-step derivation

      1. associate-*l* 0.0%

         \[\leadsto \color{blue}{d \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. *-commutative 0.0%

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d} \]

      3. unpow2 0.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      4. rem-square-sqrt 38.1%

         \[\leadsto \left(\color{blue}{-1} \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \cdot d \]

      5. *-commutative 38.1%

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot -1\right)} \cdot d \]

      6. associate-*r* 38.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      7. unpow1/2 38.1%

         \[\leadsto \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \cdot \left(-1 \cdot d\right) \]

      8. rem-exp-log 36.2%

         \[\leadsto {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \cdot \left(-1 \cdot d\right) \]

      9. exp-neg 36.2%

         \[\leadsto {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \cdot \left(-1 \cdot d\right) \]

      10. exp-prod 36.2%

          \[\leadsto \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      11. distribute-lft-neg-out 36.2%

          \[\leadsto e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \cdot \left(-1 \cdot d\right) \]

      12. distribute-rgt-neg-in 36.2%

          \[\leadsto e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \cdot \left(-1 \cdot d\right) \]

      13. metadata-eval 36.2%

          \[\leadsto e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \cdot \left(-1 \cdot d\right) \]

      14. exp-to-pow 38.1%

          \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \left(-1 \cdot d\right) \]

      15. neg-mul-1 38.1%

          \[\leadsto {\left(h \cdot \ell\right)}^{-0.5} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 38.1%

      \[\leadsto \color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot \left(-d\right)} \]

   if -1.6e-184 < l

   1. Initial program 65.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 43.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. pow1 43.0%

         \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

      2. pow1/2 43.0%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

      3. inv-pow 43.0%

         \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

      4. pow-pow 43.0%

         \[\leadsto {\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

      5. metadata-eval 43.0%

         \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

   7. Applied egg-rr 43.0%

      \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 43.0%

         \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 43.0%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{-184}:\\ \;\;\;\;d \cdot \left(-{\left(h \cdot \ell\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(h \cdot \ell\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * sqrt((1.0d0 / (h * l)))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.sqrt((1.0 / (h * l)))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.7%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 28.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Add Preprocessing

Alternative 21: 26.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot {\left(h \cdot \ell\right)}^{-0.5} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (* d (pow (* h l) -0.5)))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d * pow((h * l), -0.5);
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d * ((h * l) ** (-0.5d0))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d * Math.pow((h * l), -0.5);
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d * math.pow((h * l), -0.5)

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d * (Float64(h * l) ^ -0.5))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d * ((h * l) ^ -0.5);
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d * N[Power[N[(h * l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.7%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 28.1%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. pow1 28.1%

      \[\leadsto \color{blue}{{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)}^{1}} \]

   2. pow1/2 28.1%

      \[\leadsto {\left(d \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right)}^{1} \]

   3. inv-pow 28.1%

      \[\leadsto {\left(d \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right)}^{1} \]

   4. pow-pow 27.7%

      \[\leadsto {\left(d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right)}^{1} \]

   5. metadata-eval 27.7%

      \[\leadsto {\left(d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right)}^{1} \]

7. Applied egg-rr 27.7%

   \[\leadsto \color{blue}{{\left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 27.7%

      \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 27.7%

   \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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)))))