Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 79.0%

Time: 1.3min

Alternatives: 23

Speedup: 1.0×

• could not determine a ground truth

  1. d = 1.3660324346632248e+263
  2. h = -1.5748021873054488e-44
  3. l = 9.439601103895209e-185
  4. M = 4.697052255640403e-283
  5. D = -3.2389203051047774e+218

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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M\_m}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_0}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= l -1e+86)
     (*
      (* (sqrt (/ d h)) (/ t_0 (sqrt (- l))))
      (- 1.0 (* 0.5 (pow (* D (* (* 0.5 (/ M_m d)) (sqrt (/ h l)))) 2.0))))
     (if (<= l -4e-310)
       (*
        (* (/ t_0 (sqrt (- h))) (sqrt (/ d l)))
        (- 1.0 (* 0.5 (/ (* h (pow (* (/ D d) (* 0.5 M_m)) 2.0)) l))))
       (*
        (/ d (* (sqrt l) (sqrt h)))
        (fma (/ h l) (* -0.5 (pow (* M_m (/ D (* d 2.0))) 2.0)) 1.0))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (l <= -1e+86) {
		tmp = (sqrt((d / h)) * (t_0 / sqrt(-l))) * (1.0 - (0.5 * pow((D * ((0.5 * (M_m / d)) * sqrt((h / l)))), 2.0)));
	} else if (l <= -4e-310) {
		tmp = ((t_0 / sqrt(-h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow(((D / d) * (0.5 * M_m)), 2.0)) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * fma((h / l), (-0.5 * pow((M_m * (D / (d * 2.0))), 2.0)), 1.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -1e+86)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * (Float64(D * Float64(Float64(0.5 * Float64(M_m / d)) * sqrt(Float64(h / l)))) ^ 2.0))));
	elseif (l <= -4e-310)
		tmp = Float64(Float64(Float64(t_0 / sqrt(Float64(-h))) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * fma(Float64(h / l), Float64(-0.5 * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)), 1.0));
	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[(-d)], $MachinePrecision]}, If[LessEqual[l, -1e+86], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(D * N[(N[(0.5 * N[(M$95$m / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4e-310], N[(N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1e86

   1. Initial program 45.1%

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

   3. Simplified 41.1%

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

      1. add-sqr-sqrt 41.1%

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

      2. pow2 41.1%

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

      3. sqrt-prod 41.1%

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

      4. unpow2 41.1%

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

      5. sqrt-prod 20.1%

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

      6. add-sqr-sqrt 48.8%

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

      7. frac-times 52.9%

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

      8. associate-/l* 48.8%

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

      9. *-commutative 48.8%

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

   6. Applied egg-rr 48.8%

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

      1. associate-*r/ 52.9%

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

      2. remove-double-neg 52.9%

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

      3. distribute-frac-neg2 52.9%

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

      4. distribute-frac-neg 52.9%

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

      5. distribute-lft-neg-in 52.9%

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

      6. *-commutative 52.9%

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

      7. associate-/l* 49.8%

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

      8. associate-*l* 50.0%

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

      9. neg-mul-1 50.0%

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

      10. distribute-rgt-neg-in 50.0%

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

      11. metadata-eval 50.0%

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

      12. *-commutative 50.0%

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

      13. times-frac 50.0%

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

      14. metadata-eval 50.0%

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

   8. Simplified 50.0%

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

      1. frac-2neg 50.0%

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

      2. sqrt-div 63.0%

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

   10. Applied egg-rr 63.0%

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

   if -1e86 < l < -3.999999999999988e-310

   1. Initial program 72.7%

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

   3. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. clear-num 70.5%

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

      3. frac-times 71.5%

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

      4. *-un-lft-identity 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. associate-*r/ 74.1%

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

      2. div-inv 73.1%

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

      3. associate-*l/ 73.1%

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

      4. *-commutative 73.1%

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

      5. clear-num 74.3%

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

      6. associate-*r/ 75.3%

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

      7. frac-times 74.3%

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

      8. div-inv 74.3%

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

      9. metadata-eval 74.3%

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

   8. Applied egg-rr 74.3%

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

      1. frac-2neg 74.3%

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

      2. sqrt-div 86.3%

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

   10. Applied egg-rr 86.3%

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

   if -3.999999999999988e-310 < l

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d -1e-309)
   (*
    (sqrt (/ d l))
    (*
     (/ (sqrt (- d)) (sqrt (- h)))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (/ M_m 2.0)) 2.0))))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (fma (/ h l) (* -0.5 (pow (* M_m (/ D (* d 2.0))) 2.0)) 1.0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= -1e-309) {
		tmp = sqrt((d / l)) * ((sqrt(-d) / sqrt(-h)) * (1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m / 2.0)), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * fma((h / l), (-0.5 * pow((M_m * (D / (d * 2.0))), 2.0)), 1.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= -1e-309)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * fma(Float64(h / l), Float64(-0.5 * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)), 1.0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, -1e-309], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.000000000000002e-309

   1. Initial program 62.7%

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

   3. Simplified 60.6%

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

      1. frac-2neg 62.2%

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

      2. sqrt-div 71.9%

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

   6. Applied egg-rr 72.4%

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

   if -1.000000000000002e-309 < d

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\_m\right)}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2.1e-130)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (/ (* D (* 0.5 M_m)) d) 2.0)) l))))
   (if (<= h -1e-311)
     (/ d (- (sqrt (* l h))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (fma (/ h l) (* -0.5 (pow (* M_m (/ D (* d 2.0))) 2.0)) 1.0)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -2.1e-130) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow(((D * (0.5 * M_m)) / d), 2.0)) / l)));
	} else if (h <= -1e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * fma((h / l), (-0.5 * pow((M_m * (D / (d * 2.0))), 2.0)), 1.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -2.1e-130)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D * Float64(0.5 * M_m)) / d) ^ 2.0)) / l))));
	elseif (h <= -1e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * fma(Float64(h / l), Float64(-0.5 * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)), 1.0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -2.1e-130], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.10000000000000002e-130

   1. Initial program 68.4%

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

   3. Simplified 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. *-commutative 65.5%

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

      2. clear-num 64.5%

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

      3. frac-times 65.3%

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

      4. *-un-lft-identity 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. associate-*r/ 67.5%

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

      2. div-inv 66.7%

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

      3. associate-*l/ 66.7%

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

      4. *-commutative 66.7%

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

      5. clear-num 67.7%

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

      6. associate-*r/ 70.6%

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

      7. frac-times 67.7%

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

      8. div-inv 67.7%

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

      9. metadata-eval 67.7%

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

   8. Applied egg-rr 67.7%

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

      1. associate-*r/ 70.6%

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

      2. *-commutative 70.6%

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

   10. Applied egg-rr 70.6%

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

   if -2.10000000000000002e-130 < h < -9.99999999999948e-312

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. sqrt-prod 46.5%

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

      4. unpow2 46.5%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. frac-times 49.3%

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

      8. associate-/l* 49.2%

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

      9. *-commutative 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.3%

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

      2. remove-double-neg 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. distribute-lft-neg-in 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-/l* 49.1%

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

      8. associate-*l* 49.3%

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

      9. neg-mul-1 49.3%

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

      10. distribute-rgt-neg-in 49.3%

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

      11. metadata-eval 49.3%

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

      12. *-commutative 49.3%

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

      13. times-frac 49.3%

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

      14. metadata-eval 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. sqrt-div 53.8%

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

      3. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

   11. Taylor expanded in d around -inf 63.6%

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

   13. Simplified 63.6%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -8e-67)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M_m (/ D (* d 2.0))) 2.0)))))
   (if (<= h -1e-311)
     (* (- d) (pow (* l h) -0.5))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -8e-67) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((M_m * (D / (d * 2.0))), 2.0))));
	} else if (h <= -1e-311) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-8d-67)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m_m * (d_1 / (d * 2.0d0))) ** 2.0d0))))
    else if (h <= (-1d-311)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -8e-67) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M_m * (D / (d * 2.0))), 2.0))));
	} else if (h <= -1e-311) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -8e-67:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((M_m * (D / (d * 2.0))), 2.0))))
	elif h <= -1e-311:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -8e-67)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)))));
	elseif (h <= -1e-311)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -8e-67)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((M_m * (D / (d * 2.0))) ^ 2.0))));
	elseif (h <= -1e-311)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -8e-67], 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[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.99999999999999954e-67

   1. Initial program 65.1%

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

   3. Simplified 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. pow2 64.0%

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

      3. sqrt-prod 64.0%

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

      4. unpow2 64.0%

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

      5. sqrt-prod 32.0%

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

      6. add-sqr-sqrt 67.3%

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

      7. frac-times 68.5%

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

      8. associate-/l* 67.3%

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

      9. *-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*l* 67.4%

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

      2. *-commutative 67.4%

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

   8. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. sqrt-unprod 59.0%

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

      3. cancel-sign-sub-inv 59.0%

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

      4. metadata-eval 59.0%

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

      5. associate-*r* 59.0%

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

      6. unpow-prod-down 56.0%

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

      7. associate-/r* 56.0%

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

      8. pow2 56.0%

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

      9. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. unpow1 56.0%

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

       2. *-commutative 56.0%

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

       3. *-commutative 56.0%

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

       4. associate-/l/ 56.0%

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

   12. Simplified 56.0%

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

   if -7.99999999999999954e-67 < h < -9.99999999999948e-312

   1. Initial program 58.6%

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

   3. Simplified 54.8%

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

      1. add-sqr-sqrt 54.8%

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

      2. pow2 54.8%

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

      3. sqrt-prod 54.7%

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

      4. unpow2 54.7%

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

      5. sqrt-prod 35.9%

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

      6. add-sqr-sqrt 56.6%

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

      7. frac-times 60.5%

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

      8. associate-/l* 56.6%

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

      9. *-commutative 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*r/ 60.5%

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

      2. remove-double-neg 60.5%

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

      3. distribute-frac-neg2 60.5%

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

      4. distribute-frac-neg 60.5%

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

      5. distribute-lft-neg-in 60.5%

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

      6. *-commutative 60.5%

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

      7. associate-/l* 58.3%

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

      8. associate-*l* 58.5%

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

      9. neg-mul-1 58.5%

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

      10. distribute-rgt-neg-in 58.5%

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

      11. metadata-eval 58.5%

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

      12. *-commutative 58.5%

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

      13. times-frac 58.5%

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

      14. metadata-eval 58.5%

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

   8. Simplified 58.5%

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

      1. clear-num 58.4%

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

      2. sqrt-div 61.6%

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

      3. metadata-eval 61.6%

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

   10. Applied egg-rr 61.6%

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

   11. Taylor expanded in d around -inf 60.7%

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

   13. Simplified 60.7%

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

       1. div-inv 60.7%

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

       2. pow1/2 60.7%

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

       3. pow-flip 60.7%

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

       4. metadata-eval 60.7%

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

   15. Applied egg-rr 60.7%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -4.2 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left({\left(M\_m \cdot \frac{\frac{D}{2}}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -4.2e-67)
   (*
    (fma (pow (* M_m (/ (/ D 2.0) d)) 2.0) (* (/ h l) -0.5) 1.0)
    (sqrt (* d (/ (/ d l) h))))
   (if (<= h -1e-311)
     (* (- d) (pow (* l h) -0.5))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -4.2e-67) {
		tmp = fma(pow((M_m * ((D / 2.0) / d)), 2.0), ((h / l) * -0.5), 1.0) * sqrt((d * ((d / l) / h)));
	} else if (h <= -1e-311) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -4.2e-67)
		tmp = Float64(fma((Float64(M_m * Float64(Float64(D / 2.0) / d)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(Float64(d * Float64(Float64(d / l) / h))));
	elseif (h <= -1e-311)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -0.5))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[h, -4.2e-67], N[(N[(N[Power[N[(M$95$m * N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d * N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -4.2000000000000003e-67

   1. Initial program 65.1%

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

   3. Simplified 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. pow2 64.0%

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

      3. sqrt-prod 64.0%

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

      4. unpow2 64.0%

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

      5. sqrt-prod 32.0%

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

      6. add-sqr-sqrt 67.3%

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

      7. frac-times 68.5%

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

      8. associate-/l* 67.3%

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

      9. *-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*l* 67.4%

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

      2. *-commutative 67.4%

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

   8. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. sqrt-unprod 59.0%

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

      3. cancel-sign-sub-inv 59.0%

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

      4. metadata-eval 59.0%

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

      5. associate-*r* 59.0%

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

      6. unpow-prod-down 56.0%

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

      7. associate-/r* 56.0%

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

      8. pow2 56.0%

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

      9. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. unpow1 56.0%

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

       2. *-commutative 56.0%

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

       3. +-commutative 56.0%

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

       4. *-commutative 56.0%

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

       5. associate-*l* 56.0%

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

       6. fma-define 56.0%

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

       7. associate-*l/ 50.0%

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

       8. associate-/l* 56.2%

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

   12. Simplified 56.2%

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

   if -4.2000000000000003e-67 < h < -9.99999999999948e-312

   1. Initial program 58.6%

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

   3. Simplified 54.8%

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

      1. add-sqr-sqrt 54.8%

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

      2. pow2 54.8%

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

      3. sqrt-prod 54.7%

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

      4. unpow2 54.7%

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

      5. sqrt-prod 35.9%

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

      6. add-sqr-sqrt 56.6%

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

      7. frac-times 60.5%

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

      8. associate-/l* 56.6%

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

      9. *-commutative 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*r/ 60.5%

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

      2. remove-double-neg 60.5%

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

      3. distribute-frac-neg2 60.5%

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

      4. distribute-frac-neg 60.5%

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

      5. distribute-lft-neg-in 60.5%

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

      6. *-commutative 60.5%

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

      7. associate-/l* 58.3%

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

      8. associate-*l* 58.5%

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

      9. neg-mul-1 58.5%

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

      10. distribute-rgt-neg-in 58.5%

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

      11. metadata-eval 58.5%

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

      12. *-commutative 58.5%

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

      13. times-frac 58.5%

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

      14. metadata-eval 58.5%

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

   8. Simplified 58.5%

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

      1. clear-num 58.4%

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

      2. sqrt-div 61.6%

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

      3. metadata-eval 61.6%

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

   10. Applied egg-rr 61.6%

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

   11. Taylor expanded in d around -inf 60.7%

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

   13. Simplified 60.7%

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

       1. div-inv 60.7%

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

       2. pow1/2 60.7%

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

       3. pow-flip 60.7%

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

       4. metadata-eval 60.7%

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

   15. Applied egg-rr 60.7%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -4.2 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left({\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d \cdot \frac{\frac{d}{\ell}}{h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M\_m}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2.2e-130)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (* (/ D d) (/ M_m 2.0)) 2.0))))))
   (if (<= h -1e-311)
     (/ d (- (sqrt (* l h))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -2.2e-130) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow(((D / d) * (M_m / 2.0)), 2.0)))));
	} else if (h <= -1e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-2.2d-130)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * (((d_1 / d) * (m_m / 2.0d0)) ** 2.0d0)))))
    else if (h <= (-1d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -2.2e-130) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow(((D / d) * (M_m / 2.0)), 2.0)))));
	} else if (h <= -1e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -2.2e-130:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow(((D / d) * (M_m / 2.0)), 2.0)))))
	elif h <= -1e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -2.2e-130)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M_m / 2.0)) ^ 2.0))))));
	elseif (h <= -1e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -2.2e-130)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * (((D / d) * (M_m / 2.0)) ^ 2.0)))));
	elseif (h <= -1e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -2.2e-130], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M$95$m / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.1999999999999999e-130

   1. Initial program 68.4%

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

   3. Simplified 65.5%

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

   if -2.1999999999999999e-130 < h < -9.99999999999948e-312

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. sqrt-prod 46.5%

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

      4. unpow2 46.5%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. frac-times 49.3%

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

      8. associate-/l* 49.2%

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

      9. *-commutative 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.3%

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

      2. remove-double-neg 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. distribute-lft-neg-in 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-/l* 49.1%

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

      8. associate-*l* 49.3%

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

      9. neg-mul-1 49.3%

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

      10. distribute-rgt-neg-in 49.3%

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

      11. metadata-eval 49.3%

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

      12. *-commutative 49.3%

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

      13. times-frac 49.3%

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

      14. metadata-eval 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. sqrt-div 53.8%

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

      3. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

   11. Taylor expanded in d around -inf 63.6%

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

   13. Simplified 63.6%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.2 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M\_m}{2 \cdot \frac{d}{D}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2.1e-130)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (/ M_m (* 2.0 (/ d D))) 2.0))))))
   (if (<= h -1e-311)
     (/ d (- (sqrt (* l h))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -2.1e-130) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((M_m / (2.0 * (d / D))), 2.0)))));
	} else if (h <= -1e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-2.1d-130)) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((m_m / (2.0d0 * (d / d_1))) ** 2.0d0)))))
    else if (h <= (-1d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -2.1e-130) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((M_m / (2.0 * (d / D))), 2.0)))));
	} else if (h <= -1e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -2.1e-130:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((M_m / (2.0 * (d / D))), 2.0)))))
	elif h <= -1e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -2.1e-130)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(M_m / Float64(2.0 * Float64(d / D))) ^ 2.0))))));
	elseif (h <= -1e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -2.1e-130)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((M_m / (2.0 * (d / D))) ^ 2.0)))));
	elseif (h <= -1e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -2.1e-130], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(M$95$m / N[(2.0 * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.10000000000000002e-130

   1. Initial program 68.4%

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

   3. Simplified 65.5%

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

      1. *-commutative 65.5%

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

      2. clear-num 64.5%

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

      3. frac-times 65.3%

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

      4. *-un-lft-identity 65.3%

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

   6. Applied egg-rr 65.3%

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

   if -2.10000000000000002e-130 < h < -9.99999999999948e-312

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. sqrt-prod 46.5%

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

      4. unpow2 46.5%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. frac-times 49.3%

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

      8. associate-/l* 49.2%

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

      9. *-commutative 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.3%

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

      2. remove-double-neg 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. distribute-lft-neg-in 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-/l* 49.1%

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

      8. associate-*l* 49.3%

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

      9. neg-mul-1 49.3%

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

      10. distribute-rgt-neg-in 49.3%

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

      11. metadata-eval 49.3%

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

      12. *-commutative 49.3%

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

      13. times-frac 49.3%

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

      14. metadata-eval 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. sqrt-div 53.8%

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

      3. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

   11. Taylor expanded in d around -inf 63.6%

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

   13. Simplified 63.6%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(M\_m \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2.1e-130)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (* h (/ (pow (* D (* M_m (/ 0.5 d))) 2.0) l)))))
   (if (<= h -1e-311)
     (/ d (- (sqrt (* l h))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -2.1e-130) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * (h * (pow((D * (M_m * (0.5 / d))), 2.0) / l))));
	} else if (h <= -1e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-2.1d-130)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * (h * (((d_1 * (m_m * (0.5d0 / d))) ** 2.0d0) / l))))
    else if (h <= (-1d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -2.1e-130) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * (h * (Math.pow((D * (M_m * (0.5 / d))), 2.0) / l))));
	} else if (h <= -1e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -2.1e-130:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * (h * (math.pow((D * (M_m * (0.5 / d))), 2.0) / l))))
	elif h <= -1e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -2.1e-130)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(h * Float64((Float64(D * Float64(M_m * Float64(0.5 / d))) ^ 2.0) / l)))));
	elseif (h <= -1e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -2.1e-130)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * (h * (((D * (M_m * (0.5 / d))) ^ 2.0) / l))));
	elseif (h <= -1e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -2.1e-130], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(h * N[(N[Power[N[(D * N[(M$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.10000000000000002e-130

   1. Initial program 68.4%

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

   3. Simplified 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/ 67.7%

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

      2. add-sqr-sqrt 35.5%

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

      3. add-sqr-sqrt 67.7%

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

      4. frac-times 70.6%

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

      5. associate-/l* 67.7%

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

      6. *-commutative 67.7%

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

   6. Applied egg-rr 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/l* 66.8%

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

      3. *-rgt-identity 66.8%

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

      4. times-frac 66.8%

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

      5. metadata-eval 66.8%

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

      6. associate-*l* 66.8%

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

      7. associate-*r/ 68.8%

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

      8. associate-*l/ 68.8%

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

      9. associate-/l* 68.8%

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

      10. *-commutative 68.8%

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

      11. associate-*l* 65.4%

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

   8. Simplified 65.4%

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

   if -2.10000000000000002e-130 < h < -9.99999999999948e-312

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. sqrt-prod 46.5%

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

      4. unpow2 46.5%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. frac-times 49.3%

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

      8. associate-/l* 49.2%

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

      9. *-commutative 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.3%

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

      2. remove-double-neg 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. distribute-lft-neg-in 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-/l* 49.1%

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

      8. associate-*l* 49.3%

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

      9. neg-mul-1 49.3%

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

      10. distribute-rgt-neg-in 49.3%

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

      11. metadata-eval 49.3%

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

      12. *-commutative 49.3%

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

      13. times-frac 49.3%

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

      14. metadata-eval 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. sqrt-div 53.8%

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

      3. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

   11. Taylor expanded in d around -inf 63.6%

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

   13. Simplified 63.6%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(h \cdot \frac{{\left(D \cdot \left(M \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -2.1e-130)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (* M_m (/ D (* d 2.0))) 2.0)) l))))
   (if (<= h -1e-311)
     (/ d (- (sqrt (* l h))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -2.1e-130) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow((M_m * (D / (d * 2.0))), 2.0)) / l)));
	} else if (h <= -1e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-2.1d-130)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h * ((m_m * (d_1 / (d * 2.0d0))) ** 2.0d0)) / l)))
    else if (h <= (-1d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -2.1e-130) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h * Math.pow((M_m * (D / (d * 2.0))), 2.0)) / l)));
	} else if (h <= -1e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -2.1e-130:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h * math.pow((M_m * (D / (d * 2.0))), 2.0)) / l)))
	elif h <= -1e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -2.1e-130)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)) / l))));
	elseif (h <= -1e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -2.1e-130)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * ((M_m * (D / (d * 2.0))) ^ 2.0)) / l)));
	elseif (h <= -1e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -2.1e-130], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.10000000000000002e-130

   1. Initial program 68.4%

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

   3. Simplified 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/ 67.7%

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

      2. add-sqr-sqrt 35.5%

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

      3. add-sqr-sqrt 67.7%

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

      4. frac-times 70.6%

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

      5. associate-/l* 67.7%

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

      6. *-commutative 67.7%

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

   6. Applied egg-rr 67.7%

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

   if -2.10000000000000002e-130 < h < -9.99999999999948e-312

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. sqrt-prod 46.5%

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

      4. unpow2 46.5%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. frac-times 49.3%

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

      8. associate-/l* 49.2%

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

      9. *-commutative 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.3%

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

      2. remove-double-neg 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. distribute-lft-neg-in 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-/l* 49.1%

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

      8. associate-*l* 49.3%

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

      9. neg-mul-1 49.3%

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

      10. distribute-rgt-neg-in 49.3%

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

      11. metadata-eval 49.3%

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

      12. *-commutative 49.3%

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

      13. times-frac 49.3%

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

      14. metadata-eval 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. sqrt-div 53.8%

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

      3. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

   11. Taylor expanded in d around -inf 63.6%

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

   13. Simplified 63.6%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -7.6 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\_m\right)}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -7.6e-130)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (* 0.5 (/ (* h (pow (/ (* D (* 0.5 M_m)) d) 2.0)) l))))
   (if (<= h -1e-311)
     (/ d (- (sqrt (* l h))))
     (*
      (/ d (* (sqrt l) (sqrt h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -7.6e-130) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * pow(((D * (0.5 * M_m)) / d), 2.0)) / l)));
	} else if (h <= -1e-311) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-7.6d-130)) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - (0.5d0 * ((h * (((d_1 * (0.5d0 * m_m)) / d) ** 2.0d0)) / l)))
    else if (h <= (-1d-311)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -7.6e-130) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - (0.5 * ((h * Math.pow(((D * (0.5 * M_m)) / d), 2.0)) / l)));
	} else if (h <= -1e-311) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -7.6e-130:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - (0.5 * ((h * math.pow(((D * (0.5 * M_m)) / d), 2.0)) / l)))
	elif h <= -1e-311:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -7.6e-130)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(D * Float64(0.5 * M_m)) / d) ^ 2.0)) / l))));
	elseif (h <= -1e-311)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -7.6e-130)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - (0.5 * ((h * (((D * (0.5 * M_m)) / d) ^ 2.0)) / l)));
	elseif (h <= -1e-311)
		tmp = d / -sqrt((l * h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -7.6e-130], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(D * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.5999999999999997e-130

   1. Initial program 68.4%

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

   3. Simplified 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. *-commutative 65.5%

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

      2. clear-num 64.5%

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

      3. frac-times 65.3%

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

      4. *-un-lft-identity 65.3%

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

   6. Applied egg-rr 65.3%

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

      1. associate-*r/ 67.5%

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

      2. div-inv 66.7%

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

      3. associate-*l/ 66.7%

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

      4. *-commutative 66.7%

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

      5. clear-num 67.7%

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

      6. associate-*r/ 70.6%

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

      7. frac-times 67.7%

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

      8. div-inv 67.7%

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

      9. metadata-eval 67.7%

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

   8. Applied egg-rr 67.7%

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

      1. associate-*r/ 70.6%

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

      2. *-commutative 70.6%

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

   10. Applied egg-rr 70.6%

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

   if -7.5999999999999997e-130 < h < -9.99999999999948e-312

   1. Initial program 46.5%

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

   3. Simplified 46.5%

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

      1. add-sqr-sqrt 46.6%

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

      2. pow2 46.6%

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

      3. sqrt-prod 46.5%

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

      4. unpow2 46.5%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. frac-times 49.3%

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

      8. associate-/l* 49.2%

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

      9. *-commutative 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. associate-*r/ 49.3%

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

      2. remove-double-neg 49.3%

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

      3. distribute-frac-neg2 49.3%

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

      4. distribute-frac-neg 49.3%

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

      5. distribute-lft-neg-in 49.3%

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

      6. *-commutative 49.3%

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

      7. associate-/l* 49.1%

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

      8. associate-*l* 49.3%

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

      9. neg-mul-1 49.3%

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

      10. distribute-rgt-neg-in 49.3%

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

      11. metadata-eval 49.3%

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

      12. *-commutative 49.3%

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

      13. times-frac 49.3%

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

      14. metadata-eval 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. sqrt-div 53.8%

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

      3. metadata-eval 53.8%

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

   10. Applied egg-rr 53.8%

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

   11. Taylor expanded in d around -inf 63.6%

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

   13. Simplified 63.6%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. fma-undefine 71.5%

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

      2. associate-*r/ 71.5%

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

      3. frac-times 71.5%

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

      4. div-inv 71.5%

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

      5. metadata-eval 71.5%

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

   10. Applied egg-rr 81.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.6 \cdot 10^{-130}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\frac{D \cdot \left(0.5 \cdot M\right)}{d}\right)}^{2}}{\ell}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}\\ \mathbf{if}\;h \leq -7.5 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot t\_0\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot t\_0, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (pow (* M_m (/ D (* d 2.0))) 2.0)))
   (if (<= h -7.5e-68)
     (* (sqrt (* (/ d h) (/ d l))) (+ 1.0 (* -0.5 (* (/ h l) t_0))))
     (if (<= h -1e-311)
       (* (- d) (pow (* l h) -0.5))
       (* (fma (/ h l) (* -0.5 t_0) 1.0) (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = pow((M_m * (D / (d * 2.0))), 2.0);
	double tmp;
	if (h <= -7.5e-68) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * t_0)));
	} else if (h <= -1e-311) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = fma((h / l), (-0.5 * t_0), 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0
	tmp = 0.0
	if (h <= -7.5e-68)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * t_0))));
	elseif (h <= -1e-311)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * t_0), 1.0) * Float64(d / sqrt(Float64(l * 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[Power[N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[h, -7.5e-68], 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] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -7.50000000000000081e-68

   1. Initial program 65.1%

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

   3. Simplified 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. pow2 64.0%

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

      3. sqrt-prod 64.0%

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

      4. unpow2 64.0%

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

      5. sqrt-prod 32.0%

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

      6. add-sqr-sqrt 67.3%

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

      7. frac-times 68.5%

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

      8. associate-/l* 67.3%

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

      9. *-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*l* 67.4%

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

      2. *-commutative 67.4%

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

   8. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. sqrt-unprod 59.0%

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

      3. cancel-sign-sub-inv 59.0%

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

      4. metadata-eval 59.0%

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

      5. associate-*r* 59.0%

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

      6. unpow-prod-down 56.0%

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

      7. associate-/r* 56.0%

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

      8. pow2 56.0%

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

      9. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. unpow1 56.0%

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

       2. *-commutative 56.0%

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

       3. *-commutative 56.0%

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

       4. associate-/l/ 56.0%

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

   12. Simplified 56.0%

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

   if -7.50000000000000081e-68 < h < -9.99999999999948e-312

   1. Initial program 58.6%

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

   3. Simplified 54.8%

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

      1. add-sqr-sqrt 54.8%

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

      2. pow2 54.8%

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

      3. sqrt-prod 54.7%

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

      4. unpow2 54.7%

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

      5. sqrt-prod 35.9%

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

      6. add-sqr-sqrt 56.6%

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

      7. frac-times 60.5%

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

      8. associate-/l* 56.6%

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

      9. *-commutative 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*r/ 60.5%

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

      2. remove-double-neg 60.5%

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

      3. distribute-frac-neg2 60.5%

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

      4. distribute-frac-neg 60.5%

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

      5. distribute-lft-neg-in 60.5%

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

      6. *-commutative 60.5%

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

      7. associate-/l* 58.3%

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

      8. associate-*l* 58.5%

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

      9. neg-mul-1 58.5%

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

      10. distribute-rgt-neg-in 58.5%

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

      11. metadata-eval 58.5%

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

      12. *-commutative 58.5%

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

      13. times-frac 58.5%

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

      14. metadata-eval 58.5%

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

   8. Simplified 58.5%

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

      1. clear-num 58.4%

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

      2. sqrt-div 61.6%

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

      3. metadata-eval 61.6%

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

   10. Applied egg-rr 61.6%

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

   11. Taylor expanded in d around -inf 60.7%

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

   13. Simplified 60.7%

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

       1. div-inv 60.7%

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

       2. pow1/2 60.7%

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

       3. pow-flip 60.7%

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

       4. metadata-eval 60.7%

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

   15. Applied egg-rr 60.7%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. sqrt-unprod 71.5%

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

      2. *-commutative 71.5%

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

      3. pow1/2 71.5%

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

   10. Applied egg-rr 71.5%

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

       1. unpow1/2 71.5%

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

   12. Simplified 71.5%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -7.5 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -2.7 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (<= h -2.7e-68)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M_m (/ D (* d 2.0))) 2.0)))))
   (if (<= h -1e-311)
     (* (- d) (pow (* l h) -0.5))
     (/
      (* d (fma (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -0.5) 1.0))
      (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (h <= -2.7e-68) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((M_m * (D / (d * 2.0))), 2.0))));
	} else if (h <= -1e-311) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = (d * fma((h / l), (pow(((D / d) * (0.5 * M_m)), 2.0) * -0.5), 1.0)) / sqrt((l * h));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (h <= -2.7e-68)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)))));
	elseif (h <= -1e-311)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(Float64(d * fma(Float64(h / l), Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -0.5), 1.0)) / sqrt(Float64(l * 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[h, -2.7e-68], 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[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.7000000000000002e-68

   1. Initial program 65.1%

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

   3. Simplified 64.0%

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

      1. add-sqr-sqrt 64.0%

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

      2. pow2 64.0%

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

      3. sqrt-prod 64.0%

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

      4. unpow2 64.0%

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

      5. sqrt-prod 32.0%

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

      6. add-sqr-sqrt 67.3%

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

      7. frac-times 68.5%

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

      8. associate-/l* 67.3%

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

      9. *-commutative 67.3%

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

   6. Applied egg-rr 67.3%

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

      1. associate-*l* 67.4%

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

      2. *-commutative 67.4%

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

   8. Simplified 67.4%

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

      1. pow1 67.4%

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

      2. sqrt-unprod 59.0%

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

      3. cancel-sign-sub-inv 59.0%

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

      4. metadata-eval 59.0%

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

      5. associate-*r* 59.0%

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

      6. unpow-prod-down 56.0%

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

      7. associate-/r* 56.0%

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

      8. pow2 56.0%

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

      9. add-sqr-sqrt 56.0%

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

   10. Applied egg-rr 56.0%

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

       1. unpow1 56.0%

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

       2. *-commutative 56.0%

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

       3. *-commutative 56.0%

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

       4. associate-/l/ 56.0%

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

   12. Simplified 56.0%

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

   if -2.7000000000000002e-68 < h < -9.99999999999948e-312

   1. Initial program 58.6%

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

   3. Simplified 54.8%

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

      1. add-sqr-sqrt 54.8%

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

      2. pow2 54.8%

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

      3. sqrt-prod 54.7%

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

      4. unpow2 54.7%

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

      5. sqrt-prod 35.9%

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

      6. add-sqr-sqrt 56.6%

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

      7. frac-times 60.5%

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

      8. associate-/l* 56.6%

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

      9. *-commutative 56.6%

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

   6. Applied egg-rr 56.6%

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

      1. associate-*r/ 60.5%

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

      2. remove-double-neg 60.5%

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

      3. distribute-frac-neg2 60.5%

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

      4. distribute-frac-neg 60.5%

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

      5. distribute-lft-neg-in 60.5%

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

      6. *-commutative 60.5%

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

      7. associate-/l* 58.3%

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

      8. associate-*l* 58.5%

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

      9. neg-mul-1 58.5%

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

      10. distribute-rgt-neg-in 58.5%

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

      11. metadata-eval 58.5%

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

      12. *-commutative 58.5%

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

      13. times-frac 58.5%

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

      14. metadata-eval 58.5%

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

   8. Simplified 58.5%

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

      1. clear-num 58.4%

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

      2. sqrt-div 61.6%

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

      3. metadata-eval 61.6%

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

   10. Applied egg-rr 61.6%

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

   11. Taylor expanded in d around -inf 60.7%

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

   13. Simplified 60.7%

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

       1. div-inv 60.7%

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

       2. pow1/2 60.7%

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

       3. pow-flip 60.7%

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

       4. metadata-eval 60.7%

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

   15. Applied egg-rr 60.7%

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

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   6. Applied egg-rr 81.9%

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

      1. unpow1 81.9%

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

      2. +-commutative 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. fma-define 81.9%

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

      6. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r/ 86.0%

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

      3. frac-times 86.0%

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

      4. div-inv 86.0%

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

      5. metadata-eval 86.0%

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

      6. sqrt-unprod 73.2%

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

      7. *-commutative 73.2%

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

   10. Applied egg-rr 73.2%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -2.7 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-141}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \sqrt{e^{\frac{1}{\ell \cdot h}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

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 -4.4e-141)
   (* (- d) (pow (* l h) -0.5))
   (if (<= l 1.65e-307)
     (* d (sqrt (+ (exp (/ 1.0 (* l h))) -1.0)))
     (*
      (/ d (sqrt (* l h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 <= -4.4e-141) {
		tmp = -d * pow((l * h), -0.5);
	} else if (l <= 1.65e-307) {
		tmp = d * sqrt((exp((1.0 / (l * h))) + -1.0));
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 <= (-4.4d-141)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else if (l <= 1.65d-307) then
        tmp = d * sqrt((exp((1.0d0 / (l * h))) + (-1.0d0)))
    else
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 <= -4.4e-141) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else if (l <= 1.65e-307) {
		tmp = d * Math.sqrt((Math.exp((1.0 / (l * h))) + -1.0));
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 <= -4.4e-141:
		tmp = -d * math.pow((l * h), -0.5)
	elif l <= 1.65e-307:
		tmp = d * math.sqrt((math.exp((1.0 / (l * h))) + -1.0))
	else:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 <= -4.4e-141)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	elseif (l <= 1.65e-307)
		tmp = Float64(d * sqrt(Float64(exp(Float64(1.0 / Float64(l * h))) + -1.0)));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 <= -4.4e-141)
		tmp = -d * ((l * h) ^ -0.5);
	elseif (l <= 1.65e-307)
		tmp = d * sqrt((exp((1.0 / (l * h))) + -1.0));
	else
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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, -4.4e-141], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e-307], N[(d * N[Sqrt[N[(N[Exp[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.40000000000000018e-141

   1. Initial program 58.5%

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

   3. Simplified 55.6%

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

      1. add-sqr-sqrt 55.6%

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

      2. pow2 55.6%

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

      3. sqrt-prod 55.6%

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

      4. unpow2 55.6%

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

      5. sqrt-prod 31.5%

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

      6. add-sqr-sqrt 59.3%

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

      7. frac-times 62.2%

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

      8. associate-/l* 59.3%

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

      9. *-commutative 59.3%

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

   6. Applied egg-rr 59.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 62.2%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 58.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 58.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 58.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 58.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 58.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 60.4%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 60.4%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 60.4%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 45.3%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 45.3%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 45.3%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 45.3%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 45.3%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 45.3%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 45.3%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if -4.40000000000000018e-141 < l < 1.65e-307

   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 73.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 23.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 23.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

      2. expm1-undefine 23.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \]

      3. associate-/r* 23.1%

         \[\leadsto d \cdot \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{h}}{\ell}}\right)} - 1} \]

   7. Applied egg-rr 23.1%

      \[\leadsto d \cdot \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell}\right)} - 1}} \]

   8. Taylor expanded in h around inf 40.0%

      \[\leadsto d \cdot \sqrt{e^{\color{blue}{\frac{1}{h \cdot \ell}}} - 1} \]

   if 1.65e-307 < l

   1. Initial program 64.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 82.6%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)} \]

      2. +-commutative 82.6%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \color{blue}{\left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 82.6%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot -0.5} + 1\right) \]

      4. associate-*r* 82.6%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)} + 1\right) \]

      5. fma-define 82.6%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)} \]

      6. *-commutative 82.6%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot -0.5, 1\right) \]

   8. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right)} \]
   9. Step-by-step derivation

      1. sqrt-unprod 72.1%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

      2. *-commutative 72.1%

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

      3. pow1/2 72.1%

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

   10. Applied egg-rr 72.1%

       \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]
   11. Step-by-step derivation

       1. unpow1/2 72.1%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

   12. Simplified 72.1%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]
   13. Step-by-step derivation

       1. fma-undefine 72.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]

       2. associate-*r/ 72.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot -0.5\right) + 1\right) \]

       3. frac-times 72.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot -0.5\right) + 1\right) \]

       4. div-inv 72.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right) + 1\right) \]

       5. metadata-eval 72.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right) + 1\right) \]

   14. Applied egg-rr 72.1%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{-141}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-307}:\\ \;\;\;\;d \cdot \sqrt{e^{\frac{1}{\ell \cdot h}} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M\_m \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\_m\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= h -6e-68)
   (*
    (sqrt (* (/ d h) (/ d l)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M_m (/ D (* d 2.0))) 2.0)))))
   (if (<= h -1e-311)
     (* (- d) (pow (* l h) -0.5))
     (*
      (/ d (sqrt (* l h)))
      (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (* 0.5 M_m)) 2.0) -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 (h <= -6e-68) {
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * pow((M_m * (D / (d * 2.0))), 2.0))));
	} else if (h <= -1e-311) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * (pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= (-6d-68)) then
        tmp = sqrt(((d / h) * (d / l))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m_m * (d_1 / (d * 2.0d0))) ** 2.0d0))))
    else if (h <= (-1d-311)) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = (d / sqrt((l * h))) * (1.0d0 + ((h / l) * ((((d_1 / d) * (0.5d0 * m_m)) ** 2.0d0) * (-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 (h <= -6e-68) {
		tmp = Math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M_m * (D / (d * 2.0))), 2.0))));
	} else if (h <= -1e-311) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + ((h / l) * (Math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 h <= -6e-68:
		tmp = math.sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * math.pow((M_m * (D / (d * 2.0))), 2.0))))
	elif h <= -1e-311:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = (d / math.sqrt((l * h))) * (1.0 + ((h / l) * (math.pow(((D / d) * (0.5 * M_m)), 2.0) * -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 (h <= -6e-68)
		tmp = Float64(sqrt(Float64(Float64(d / h) * Float64(d / l))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M_m * Float64(D / Float64(d * 2.0))) ^ 2.0)))));
	elseif (h <= -1e-311)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(0.5 * M_m)) ^ 2.0) * -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 (h <= -6e-68)
		tmp = sqrt(((d / h) * (d / l))) * (1.0 + (-0.5 * ((h / l) * ((M_m * (D / (d * 2.0))) ^ 2.0))));
	elseif (h <= -1e-311)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = (d / sqrt((l * h))) * (1.0 + ((h / l) * ((((D / d) * (0.5 * M_m)) ^ 2.0) * -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[h, -6e-68], 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[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-311], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(0.5 * M$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -6e-68

   1. Initial program 65.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 64.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 32.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 68.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 67.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 67.3%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*l* 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(M \cdot \left(\frac{D}{d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      2. *-commutative 67.4%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(M \cdot \left(\frac{D}{\color{blue}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 67.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. pow1 67.4%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)}^{1}} \]

      2. sqrt-unprod 59.0%

         \[\leadsto {\left(\color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \left(1 - 0.5 \cdot {\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)}^{1} \]

      3. cancel-sign-sub-inv 59.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot {\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)}\right)}^{1} \]

      4. metadata-eval 59.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot {\left(M \cdot \left(\frac{D}{2 \cdot d} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right)\right)}^{1} \]

      5. associate-*r* 59.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot {\color{blue}{\left(\left(M \cdot \frac{D}{2 \cdot d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right)\right)}^{1} \]

      6. unpow-prod-down 56.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{2}\right)}\right)\right)}^{1} \]

      7. associate-/r* 56.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \color{blue}{\frac{\frac{D}{2}}{d}}\right)}^{2} \cdot {\left(\sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\right)}^{1} \]

      8. pow2 56.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\frac{h}{\ell}} \cdot \sqrt{\frac{h}{\ell}}\right)}\right)\right)\right)}^{1} \]

      9. add-sqr-sqrt 56.0%

         \[\leadsto {\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2} \cdot \color{blue}{\frac{h}{\ell}}\right)\right)\right)}^{1} \]

   10. Applied egg-rr 56.0%

       \[\leadsto \color{blue}{{\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)}^{1}} \]
   11. Step-by-step derivation

       1. unpow1 56.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

       2. *-commutative 56.0%

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

       3. *-commutative 56.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)}\right) \]

       4. associate-/l/ 56.0%

          \[\leadsto \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \color{blue}{\frac{D}{d \cdot 2}}\right)}^{2}\right)\right) \]

   12. Simplified 56.0%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)} \]

   if -6e-68 < h < -9.99999999999948e-312

   1. Initial program 58.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 54.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 54.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 54.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 54.7%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 35.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 56.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 56.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 58.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.5%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 58.5%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 58.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 58.4%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 61.6%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 61.6%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 61.6%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 60.7%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 60.7%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 60.7%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 60.7%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 60.7%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 60.7%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 60.7%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if -9.99999999999948e-312 < h

   1. Initial program 64.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{{\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 81.9%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)} \]

      2. +-commutative 81.9%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \color{blue}{\left(-0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) + 1\right)} \]

      3. *-commutative 81.9%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\color{blue}{\left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right) \cdot -0.5} + 1\right) \]

      4. associate-*r* 81.9%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(\color{blue}{\frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot -0.5\right)} + 1\right) \]

      5. fma-define 81.9%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \color{blue}{\mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2} \cdot -0.5, 1\right)} \]

      6. *-commutative 81.9%

         \[\leadsto \frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{\color{blue}{2 \cdot d}}\right)}^{2} \cdot -0.5, 1\right) \]

   8. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right)} \]
   9. Step-by-step derivation

      1. sqrt-unprod 71.5%

         \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

      2. *-commutative 71.5%

         \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

      3. pow1/2 71.5%

         \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

   10. Applied egg-rr 71.5%

       \[\leadsto \frac{d}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]
   11. Step-by-step derivation

       1. unpow1/2 71.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]

   12. Simplified 71.5%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5, 1\right) \]
   13. Step-by-step derivation

       1. fma-undefine 71.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]

       2. associate-*r/ 71.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot -0.5\right) + 1\right) \]

       3. frac-times 71.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot -0.5\right) + 1\right) \]

       4. div-inv 71.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right) + 1\right) \]

       5. metadata-eval 71.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right) + 1\right) \]

   14. Applied egg-rr 71.5%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{h}{\ell} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \left(0.5 \cdot M\right)\right)}^{2} \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

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 (<= d 2.5e-183) (* (- d) (pow (* l h) -0.5)) (/ d (* (sqrt l) (sqrt h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 2.5e-183) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 2.5d-183) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 2.5e-183) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 2.5e-183:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 2.5e-183)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 2.5e-183)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 2.5e-183], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.5000000000000001e-183

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 36.5%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 36.5%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 36.5%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 36.5%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 36.5%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 2.5000000000000001e-183 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. sqrt-div 46.1%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}} \]

      2. metadata-eval 46.1%

         \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}} \]

      3. *-commutative 46.1%

         \[\leadsto d \cdot \frac{1}{\sqrt{\color{blue}{\ell \cdot h}}} \]

      4. sqrt-unprod 53.0%

         \[\leadsto d \cdot \frac{1}{\color{blue}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      5. div-inv 53.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}} \]

      6. associate-/r* 52.0%

         \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]

   7. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}} \]
   8. Step-by-step derivation

      1. associate-/l/ 53.0%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]

   9. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{if}\;d \leq 6.2 \cdot 10^{-184}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

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 (* l h) -0.5)))
   (if (<= d 6.2e-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((l * h), -0.5);
	double tmp;
	if (d <= 6.2e-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 = (l * h) ** (-0.5d0)
    if (d <= 6.2d-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((l * h), -0.5);
	double tmp;
	if (d <= 6.2e-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((l * h), -0.5)
	tmp = 0
	if d <= 6.2e-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(l * h) ^ -0.5
	tmp = 0.0
	if (d <= 6.2e-184)
		tmp = Float64(Float64(-d) * 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 = (l * h) ^ -0.5;
	tmp = 0.0;
	if (d <= 6.2e-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[(l * h), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[d, 6.2e-184], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 6.2000000000000004e-184

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 36.5%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 36.5%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 36.5%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 36.5%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 36.5%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 6.2000000000000004e-184 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 39.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 73.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 73.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 46.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 46.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 46.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 46.1%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 45.9%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 45.9%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 46.1%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.2 \cdot 10^{-184}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.85 \cdot 10^{-184}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

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 (<= d 1.85e-184)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ 1.0 (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1.85e-184) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 1.85d-184) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt((1.0d0 / (l * h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 1.85e-184) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 1.85e-184:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 1.85e-184)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 1.85e-184)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 1.85e-184], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.8499999999999999e-184

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 36.5%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 36.5%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 36.5%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 36.5%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 36.5%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 1.8499999999999999e-184 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.85 \cdot 10^{-184}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 7e-183) (* (- d) (pow (* l h) -0.5)) (* 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) {
	double tmp;
	if (d <= 7e-183) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 7d-183) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / h) / l))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 7e-183) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / h) / l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 7e-183:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / 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 (d <= 7e-183)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 7e-183)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / h) / l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 7e-183], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 6.99999999999999983e-183

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 36.5%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 36.5%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 36.5%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 36.5%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 36.5%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 6.99999999999999983e-183 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 44.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

      2. expm1-undefine 28.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \]

      3. associate-/r* 28.2%

         \[\leadsto d \cdot \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{h}}{\ell}}\right)} - 1} \]

   7. Applied egg-rr 28.2%

      \[\leadsto d \cdot \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell}\right)} - 1}} \]
   8. Step-by-step derivation

      1. expm1-define 45.3%

         \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell}\right)\right)}} \]

      2. expm1-log1p-u 47.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]

   9. Applied egg-rr 47.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{h}}{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 7 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<= d 2.5e-183)
   (* (- d) (pow (* l h) -0.5))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 2.5e-183) {
		tmp = -d * pow((l * h), -0.5);
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 2.5d-183) then
        tmp = -d * ((l * h) ** (-0.5d0))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (d <= 2.5e-183) {
		tmp = -d * Math.pow((l * h), -0.5);
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if d <= 2.5e-183:
		tmp = -d * math.pow((l * h), -0.5)
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (d <= 2.5e-183)
		tmp = Float64(Float64(-d) * (Float64(l * h) ^ -0.5));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (d <= 2.5e-183)
		tmp = -d * ((l * h) ^ -0.5);
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[d, 2.5e-183], N[((-d) * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.5000000000000001e-183

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. div-inv 36.5%

          \[\leadsto -\color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

       2. pow1/2 36.5%

          \[\leadsto -d \cdot \frac{1}{\color{blue}{{\left(h \cdot \ell\right)}^{0.5}}} \]

       3. pow-flip 36.5%

          \[\leadsto -d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-0.5\right)}} \]

       4. metadata-eval 36.5%

          \[\leadsto -d \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}} \]

   15. Applied egg-rr 36.5%

       \[\leadsto -\color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

   if 2.5000000000000001e-183 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-commutative 46.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 47.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   7. Simplified 47.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2.5 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq 6.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{t\_0}\\ \end{array} \end{array} \]

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 (* l h)))) (if (<= d 6.8e-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 = sqrt((l * h));
	double tmp;
	if (d <= 6.8e-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 = sqrt((l * h))
    if (d <= 6.8d-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.sqrt((l * h));
	double tmp;
	if (d <= 6.8e-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.sqrt((l * h))
	tmp = 0
	if d <= 6.8e-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 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= 6.8e-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 = sqrt((l * h));
	tmp = 0.0;
	if (d <= 6.8e-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[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, 6.8e-184], N[(d / (-t$95$0)), $MachinePrecision], N[(d / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < 6.80000000000000008e-184

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if 6.80000000000000008e-184 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 39.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 73.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 73.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 46.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 46.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 46.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 46.1%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 45.9%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 45.9%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 46.1%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

       8. rem-exp-log 42.6%

          \[\leadsto \color{blue}{e^{\log d}} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

       9. rem-exp-log 42.4%

          \[\leadsto e^{\log d} \cdot \color{blue}{e^{\log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

       10. exp-sum 41.9%

           \[\leadsto \color{blue}{e^{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

       11. log-pow 41.9%

           \[\leadsto e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}} \]

       12. metadata-eval 41.9%

           \[\leadsto e^{\log d + \color{blue}{\left(0.5 \cdot -1\right)} \cdot \log \left(h \cdot \ell\right)} \]

       13. associate-*r* 41.9%

           \[\leadsto e^{\log d + \color{blue}{0.5 \cdot \left(-1 \cdot \log \left(h \cdot \ell\right)\right)}} \]

       14. neg-mul-1 41.9%

           \[\leadsto e^{\log d + 0.5 \cdot \color{blue}{\left(-\log \left(h \cdot \ell\right)\right)}} \]

       15. distribute-rgt-neg-out 41.9%

           \[\leadsto e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}} \]

       16. log-pow 41.9%

           \[\leadsto e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)} \]

       17. unpow1/2 41.9%

           \[\leadsto e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)} \]

       18. sub-neg 41.9%

           \[\leadsto e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}} \]

       19. log-div 42.6%

           \[\leadsto e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}} \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 6.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.1 \cdot 10^{-182}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (<= d 1.1e-182) (/ d (- (sqrt (* l h)))) (* d (pow (* l 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 (d <= 1.1e-182) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * pow((l * 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 (d <= 1.1d-182) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * ((l * 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 (d <= 1.1e-182) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.pow((l * 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 d <= 1.1e-182:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.pow((l * 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 (d <= 1.1e-182)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * (Float64(l * 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 (d <= 1.1e-182)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * ((l * 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[d, 1.1e-182], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.1e-182

   1. Initial program 57.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 55.3%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 31.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 57.6%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 57.6%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 60.1%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 57.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 58.0%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 58.0%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 57.9%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 57.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]
   9. Step-by-step derivation

      1. clear-num 57.9%

         \[\leadsto \left(\sqrt{\color{blue}{\frac{1}{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      2. sqrt-div 58.9%

         \[\leadsto \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      3. metadata-eval 58.9%

         \[\leadsto \left(\frac{\color{blue}{1}}{\sqrt{\frac{h}{d}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. Applied egg-rr 58.9%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{\frac{h}{d}}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. Taylor expanded in d around -inf 36.1%

       \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   13. Simplified 36.5%

       \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

   if 1.1e-182 < d

   1. Initial program 73.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

      2. pow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. unpow2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. sqrt-prod 39.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. add-sqr-sqrt 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. frac-times 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-/l* 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. *-commutative 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. Applied egg-rr 73.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. remove-double-neg 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. distribute-frac-neg2 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. distribute-frac-neg 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. distribute-lft-neg-in 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. *-commutative 73.9%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. associate-/l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. associate-*l* 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

      9. neg-mul-1 73.8%

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      10. distribute-rgt-neg-in 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      11. metadata-eval 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      12. *-commutative 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      13. times-frac 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

      14. metadata-eval 73.8%

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   8. Simplified 73.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

   9. Taylor expanded in d around inf 46.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. unpow-1 46.2%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

       2. metadata-eval 46.2%

          \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 46.1%

          \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

       4. rem-sqrt-square 46.1%

          \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       5. rem-square-sqrt 45.9%

          \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

       6. fabs-sqr 45.9%

          \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

       7. rem-square-sqrt 46.1%

          \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   11. Simplified 46.1%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.1 \cdot 10^{-182}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.2%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right) \]

   2. pow2 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

   3. sqrt-prod 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

   4. unpow2 62.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\sqrt{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. sqrt-prod 34.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. add-sqr-sqrt 63.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. frac-times 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. associate-/l* 63.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. *-commutative 63.7%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\left(M \cdot \frac{D}{\color{blue}{d \cdot 2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

6. Applied egg-rr 63.7%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(M \cdot \frac{D}{d \cdot 2}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   2. remove-double-neg 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{M \cdot D}{\color{blue}{-\left(-d \cdot 2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. distribute-frac-neg2 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(-\frac{M \cdot D}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   4. distribute-frac-neg 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\frac{-M \cdot D}{-d \cdot 2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. distribute-lft-neg-in 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{\left(-M\right) \cdot D}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   6. *-commutative 65.2%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\frac{\color{blue}{D \cdot \left(-M\right)}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. associate-/l* 63.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(D \cdot \frac{-M}{-d \cdot 2}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   8. associate-*l* 63.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(D \cdot \left(\frac{-M}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}}^{2}\right) \]

   9. neg-mul-1 63.8%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{\color{blue}{-1 \cdot M}}{-d \cdot 2} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   10. distribute-rgt-neg-in 63.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{d \cdot \left(-2\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   11. metadata-eval 63.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{d \cdot \color{blue}{-2}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   12. *-commutative 63.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\frac{-1 \cdot M}{\color{blue}{-2 \cdot d}} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   13. times-frac 63.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\color{blue}{\left(\frac{-1}{-2} \cdot \frac{M}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

   14. metadata-eval 63.8%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot {\left(D \cdot \left(\left(\color{blue}{0.5} \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}\right) \]

8. Simplified 63.8%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(D \cdot \left(\left(0.5 \cdot \frac{M}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)\right)}^{2}}\right) \]

9. Taylor expanded in d around inf 22.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. unpow-1 22.5%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-1}}} \]

    2. metadata-eval 22.5%

       \[\leadsto d \cdot \sqrt{{\left(h \cdot \ell\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    3. pow-sqr 22.5%

       \[\leadsto d \cdot \sqrt{\color{blue}{{\left(h \cdot \ell\right)}^{-0.5} \cdot {\left(h \cdot \ell\right)}^{-0.5}}} \]

    4. rem-sqrt-square 22.1%

       \[\leadsto d \cdot \color{blue}{\left|{\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    5. rem-square-sqrt 22.0%

       \[\leadsto d \cdot \left|\color{blue}{\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}}\right| \]

    6. fabs-sqr 22.0%

       \[\leadsto d \cdot \color{blue}{\left(\sqrt{{\left(h \cdot \ell\right)}^{-0.5}} \cdot \sqrt{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

    7. rem-square-sqrt 22.1%

       \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

    8. rem-exp-log 16.4%

       \[\leadsto \color{blue}{e^{\log d}} \cdot {\left(h \cdot \ell\right)}^{-0.5} \]

    9. rem-exp-log 16.3%

       \[\leadsto e^{\log d} \cdot \color{blue}{e^{\log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

    10. exp-sum 16.1%

        \[\leadsto \color{blue}{e^{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

    11. log-pow 16.1%

        \[\leadsto e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}} \]

    12. metadata-eval 16.1%

        \[\leadsto e^{\log d + \color{blue}{\left(0.5 \cdot -1\right)} \cdot \log \left(h \cdot \ell\right)} \]

    13. associate-*r* 16.1%

        \[\leadsto e^{\log d + \color{blue}{0.5 \cdot \left(-1 \cdot \log \left(h \cdot \ell\right)\right)}} \]

    14. neg-mul-1 16.1%

        \[\leadsto e^{\log d + 0.5 \cdot \color{blue}{\left(-\log \left(h \cdot \ell\right)\right)}} \]

    15. distribute-rgt-neg-out 16.1%

        \[\leadsto e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}} \]

    16. log-pow 16.1%

        \[\leadsto e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)} \]

    17. unpow1/2 16.1%

        \[\leadsto e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)} \]

    18. sub-neg 16.1%

        \[\leadsto e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}} \]

    19. log-div 17.4%

        \[\leadsto e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}} \]

11. Simplified 22.1%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

12. Final simplification 22.1%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
13. Add Preprocessing

Alternative 23: 4.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ d \cdot \sqrt{0} \end{array} \]

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 0.0)))

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) {
	return d * sqrt(0.0);
}

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(0.0d0)
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(0.0);
}

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(0.0)

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(0.0))
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(0.0);
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[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 62.2%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 22.5%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 21.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)\right)}} \]

   2. expm1-undefine 15.8%

      \[\leadsto d \cdot \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{h \cdot \ell}\right)} - 1}} \]

   3. associate-/r* 15.8%

      \[\leadsto d \cdot \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{h}}{\ell}}\right)} - 1} \]

7. Applied egg-rr 15.8%

   \[\leadsto d \cdot \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{h}}{\ell}\right)} - 1}} \]

8. Taylor expanded in h around inf 3.7%

   \[\leadsto d \cdot \sqrt{\color{blue}{1} - 1} \]

9. Final simplification 3.7%

   \[\leadsto d \cdot \sqrt{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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)))))