Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.5% → 77.2%

Time: 16.1s

Alternatives: 22

Speedup: 3.7×

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: 66.5% 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: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -3.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{t\_1}{\sqrt{-h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(t\_0, {\left(\frac{d}{D\_m} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot t\_1}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{d} \cdot \mathsf{fma}\left({\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2} \cdot 0.25, t\_0, 1\right)}{\sqrt{\ell}} \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)) (t_1 (sqrt (- d))))
   (if (<= h -3.5e+58)
     (*
      (fma
       (* (* M (* D_m (/ 0.5 d))) -0.5)
       (/ (* (* D_m 0.5) (* M h)) (* l d))
       1.0)
      (* (sqrt (/ d l)) (/ t_1 (sqrt (- h)))))
     (if (<= h -1e-309)
       (/
        (*
         (* (sqrt (/ d h)) (fma t_0 (pow (* (/ d D_m) (/ 2.0 M)) -2.0) 1.0))
         t_1)
        (sqrt (- l)))
       (/
        (*
         (/
          (* (sqrt d) (fma (* (pow (/ (/ d D_m) M) -2.0) 0.25) t_0 1.0))
          (sqrt l))
         (sqrt d))
        (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (h / l) * -0.5;
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -3.5e+58) {
		tmp = fma(((M * (D_m * (0.5 / d))) * -0.5), (((D_m * 0.5) * (M * h)) / (l * d)), 1.0) * (sqrt((d / l)) * (t_1 / sqrt(-h)));
	} else if (h <= -1e-309) {
		tmp = ((sqrt((d / h)) * fma(t_0, pow(((d / D_m) * (2.0 / M)), -2.0), 1.0)) * t_1) / sqrt(-l);
	} else {
		tmp = (((sqrt(d) * fma((pow(((d / D_m) / M), -2.0) * 0.25), t_0, 1.0)) / sqrt(l)) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(h / l) * -0.5)
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -3.5e+58)
		tmp = Float64(fma(Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5), Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0) * Float64(sqrt(Float64(d / l)) * Float64(t_1 / sqrt(Float64(-h)))));
	elseif (h <= -1e-309)
		tmp = Float64(Float64(Float64(sqrt(Float64(d / h)) * fma(t_0, (Float64(Float64(d / D_m) * Float64(2.0 / M)) ^ -2.0), 1.0)) * t_1) / sqrt(Float64(-l)));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(d) * fma(Float64((Float64(Float64(d / D_m) / M) ^ -2.0) * 0.25), t_0, 1.0)) / sqrt(l)) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -3.5e+58], N[(N[(N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-309], N[(N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] * N[(2.0 / M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[(N[(N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.4999999999999997e58

   1. Initial program 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 61.5

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

   8. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

   if -3.4999999999999997e58 < h < -1.000000000000002e-309

   1. Initial program 82.6%

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

   4. Applied rewrites 89.3%

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

   if -1.000000000000002e-309 < h

   1. Initial program 69.8%

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

   4. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 83.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{\ell}}} \cdot \sqrt{d}}{\sqrt{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{d}{D} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{d} \cdot \mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{\ell}} \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(1 - \left({\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D\_m}{d}, \left(\left(M \cdot -0.5\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right)\right) \cdot \left(\left(\frac{h}{\ell} \cdot M\right) \cdot 0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (* (pow (/ (* M D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_0 -4e-130)
     (*
      (fma
       (/ D_m d)
       (* (* (* M -0.5) (* (/ D_m d) 0.5)) (* (* (/ h l) M) 0.5))
       1.0)
      (sqrt (* (/ (/ d l) h) d)))
     (if (<= t_0 2e-202)
       (/ (/ (- d) (sqrt (- h))) (sqrt (- l)))
       (* (sqrt (/ d h)) (sqrt (/ d l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (1.0 - ((pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= -4e-130) {
		tmp = fma((D_m / d), (((M * -0.5) * ((D_m / d) * 0.5)) * (((h / l) * M) * 0.5)), 1.0) * sqrt((((d / l) / h) * d));
	} else if (t_0 <= 2e-202) {
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_0 <= -4e-130)
		tmp = Float64(fma(Float64(D_m / d), Float64(Float64(Float64(M * -0.5) * Float64(Float64(D_m / d) * 0.5)) * Float64(Float64(Float64(h / l) * M) * 0.5)), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (t_0 <= 2e-202)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-h))) / sqrt(Float64(-l)));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-130], N[(N[(N[(D$95$m / d), $MachinePrecision] * N[(N[(N[(M * -0.5), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-202], N[(N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.0000000000000003e-130

   1. Initial program 89.9%

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

   4. Applied rewrites 79.1%

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

   6. Applied rewrites 78.0%

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

   if -4.0000000000000003e-130 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e-202

   1. Initial program 24.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

   7. Applied rewrites 41.6%

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

   9. Applied rewrites 53.5%

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

   if 2.0000000000000001e-202 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.9%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.2

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

   5. Applied rewrites 31.2%

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

   7. Applied rewrites 72.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{D}{d}, \left(\left(M \cdot -0.5\right) \cdot \left(\frac{D}{d} \cdot 0.5\right)\right) \cdot \left(\left(\frac{h}{\ell} \cdot M\right) \cdot 0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m}{d} \cdot 0.5\\ t_1 := \left(1 - \left({\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot M, \left(t\_0 \cdot M\right) \cdot \left(t\_0 \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ D_m d) 0.5))
        (t_1
         (*
          (- 1.0 (* (* (pow (/ (* M D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -4e-130)
     (*
      (fma (* (/ h l) M) (* (* t_0 M) (* t_0 -0.5)) 1.0)
      (sqrt (* (/ (/ d l) h) d)))
     (if (<= t_1 2e-202)
       (/ (/ (- d) (sqrt (- h))) (sqrt (- l)))
       (* (sqrt (/ d h)) (sqrt (/ d l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (D_m / d) * 0.5;
	double t_1 = (1.0 - ((pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -4e-130) {
		tmp = fma(((h / l) * M), ((t_0 * M) * (t_0 * -0.5)), 1.0) * sqrt((((d / l) / h) * d));
	} else if (t_1 <= 2e-202) {
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(D_m / d) * 0.5)
	t_1 = Float64(Float64(1.0 - Float64(Float64((Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -4e-130)
		tmp = Float64(fma(Float64(Float64(h / l) * M), Float64(Float64(t_0 * M) * Float64(t_0 * -0.5)), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (t_1 <= 2e-202)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-h))) / sqrt(Float64(-l)));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-130], N[(N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(t$95$0 * M), $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-202], N[(N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.0000000000000003e-130

   1. Initial program 89.9%

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

   4. Applied rewrites 79.1%

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

   6. Applied rewrites 78.0%

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

   if -4.0000000000000003e-130 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.0000000000000001e-202

   1. Initial program 24.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 24.8

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

   5. Applied rewrites 24.8%

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

   7. Applied rewrites 41.6%

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

   9. Applied rewrites 53.5%

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

   if 2.0000000000000001e-202 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 65.9%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.2

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

   5. Applied rewrites 31.2%

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

   7. Applied rewrites 72.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell} \cdot M, \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot M\right) \cdot \left(\left(\frac{D}{d} \cdot 0.5\right) \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 87.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 86.8

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

   6. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   4. Applied rewrites 4.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125}{d}, \frac{D \cdot D}{d}, \ell\right)}{\ell}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

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

Alternative 5: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{M \cdot D\_m}{d}, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \cdot \frac{\mathsf{fma}\left(\frac{-0.125 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}, \frac{D\_m \cdot D\_m}{d}, \ell\right)}{\ell}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* M D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      INFINITY)
   (*
    (fma (* -0.25 (/ (* M D_m) d)) (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0)
    (* (sqrt (/ d h)) (sqrt (/ d l))))
   (*
    (sqrt (* (/ (/ d l) h) d))
    (/ (fma (/ (* -0.125 (* (* M M) h)) d) (/ (* D_m D_m) d) l) l))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= ((double) INFINITY)) {
		tmp = fma((-0.25 * ((M * D_m) / d)), (((h / l) * M) * ((D_m / d) * 0.5)), 1.0) * (sqrt((d / h)) * sqrt((d / l)));
	} else {
		tmp = sqrt((((d / l) / h) * d)) * (fma(((-0.125 * ((M * M) * h)) / d), ((D_m * D_m) / d), l) / l);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= Inf)
		tmp = Float64(fma(Float64(-0.25 * Float64(Float64(M * D_m) / d)), Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0) * Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(d / l) / h) * d)) * Float64(fma(Float64(Float64(-0.125 * Float64(Float64(M * M) * h)) / d), Float64(Float64(D_m * D_m) / d), l) / l));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(-0.25 * N[(N[(M * D$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(-0.125 * N[(N[(M * M), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] + l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 87.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 86.8

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

   6. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 86.8

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in d around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 86.4

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

   11. Applied rewrites 86.4%

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

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 0.0%

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

   4. Applied rewrites 4.3%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 27.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(M \cdot M\right) \cdot h\right) \cdot -0.125}{d}, \frac{D \cdot D}{d}, \ell\right)}{\ell}} \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot \frac{M \cdot D}{d}, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d} \cdot \frac{\mathsf{fma}\left(\frac{-0.125 \cdot \left(\left(M \cdot M\right) \cdot h\right)}{d}, \frac{D \cdot D}{d}, \ell\right)}{\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}}}{\ell} \cdot \left(\left(\frac{D\_m \cdot D\_m}{d} \cdot -0.125\right) \cdot \left(M \cdot M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* M D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      0.0)
   (* (/ (sqrt (/ h l)) l) (* (* (/ (* D_m D_m) d) -0.125) (* M M)))
   (* (sqrt (/ d h)) (sqrt (/ d l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 0.0) {
		tmp = (sqrt((h / l)) / l) * ((((D_m * D_m) / d) * -0.125) * (M * M));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((1.0d0 - (((((m * d_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= 0.0d0) then
        tmp = (sqrt((h / l)) / l) * ((((d_m * d_m) / d) * (-0.125d0)) * (m * m))
    else
        tmp = sqrt((d / h)) * sqrt((d / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((Math.pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= 0.0) {
		tmp = (Math.sqrt((h / l)) / l) * ((((D_m * D_m) / d) * -0.125) * (M * M));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((1.0 - ((math.pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= 0.0:
		tmp = (math.sqrt((h / l)) / l) * ((((D_m * D_m) / d) * -0.125) * (M * M))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	return tmp

Julia

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

MATLAB

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

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * -0.125), $MachinePrecision] * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.9%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 16.6

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{\left({D}^{2} \cdot {M}^{2}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}}{d}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left({D}^{2} \cdot {M}^{2}\right)}}{d} \]

      3. associate-*r* N/A

         \[\leadsto \frac{-1}{8} \cdot \frac{\color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}\right) \cdot {M}^{2}}}{d} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{-1}{8} \cdot \color{blue}{\left(\frac{\sqrt{\frac{h}{{\ell}^{3}}} \cdot {D}^{2}}{d} \cdot {M}^{2}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\color{blue}{\left(\sqrt{\frac{h}{{\ell}^{3}}} \cdot \frac{{D}^{2}}{d}\right)} \cdot {M}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\color{blue}{\left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \cdot {M}^{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right) \cdot {M}^{2}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{{M}^{2} \cdot \left(\frac{-1}{8} \cdot \left(\frac{{D}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)\right)} \]

      9. associate-*r* N/A

         \[\leadsto {M}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2}}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left({M}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2}}{d}\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

   8. Applied rewrites 26.0%

      \[\leadsto \color{blue}{\left(\left(M \cdot M\right) \cdot \left(\frac{D \cdot D}{d} \cdot -0.125\right)\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 64.2%

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

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

   1. Initial program 66.0%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 31.0

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

   5. Applied rewrites 31.0%

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

   7. Applied rewrites 72.4%

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

4. Final simplification 69.2%

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

Alternative 7: 44.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (* (pow (/ (* M D_m) (* 2.0 d)) 2.0) (/ 1.0 2.0)) (/ h l)))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -4e-130)
   (/ d (sqrt (sqrt (* (* l h) (* l h)))))
   (* (sqrt (/ d h)) (sqrt (/ d l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -4e-130) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((1.0d0 - (((((m * d_m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)) * (h / l))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-4d-130)) then
        tmp = d / sqrt(sqrt(((l * h) * (l * h))))
    else
        tmp = sqrt((d / h)) * sqrt((d / l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((1.0 - ((Math.pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -4e-130) {
		tmp = d / Math.sqrt(Math.sqrt(((l * h) * (l * h))));
	} else {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((1.0 - ((math.pow(((M * D_m) / (2.0 * d)), 2.0) * (1.0 / 2.0)) * (h / l))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -4e-130:
		tmp = d / math.sqrt(math.sqrt(((l * h) * (l * h))))
	else:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64((Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)) * Float64(h / l))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -4e-130)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (((1.0 - (((((M * D_m) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)) * (h / l))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -4e-130)
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	else
		tmp = sqrt((d / h)) * sqrt((d / l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(1.0 - N[(N[(N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-130], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.0000000000000003e-130

   1. Initial program 89.9%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 15.0

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

   5. Applied rewrites 15.0%

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

   7. Applied rewrites 15.0%

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

   9. Applied rewrites 20.7%

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

   if -4.0000000000000003e-130 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 62.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 30.7

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

   5. Applied rewrites 30.7%

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

   7. Applied rewrites 68.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;h \leq -3.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{t\_1}{\sqrt{-h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(t\_0, {\left(\frac{d}{D\_m} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot t\_1}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2} \cdot 0.25, t\_0, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)) (t_1 (sqrt (- d))))
   (if (<= h -3.5e+58)
     (*
      (fma
       (* (* M (* D_m (/ 0.5 d))) -0.5)
       (/ (* (* D_m 0.5) (* M h)) (* l d))
       1.0)
      (* (sqrt (/ d l)) (/ t_1 (sqrt (- h)))))
     (if (<= h -1e-309)
       (/
        (*
         (* (sqrt (/ d h)) (fma t_0 (pow (* (/ d D_m) (/ 2.0 M)) -2.0) 1.0))
         t_1)
        (sqrt (- l)))
       (/
        (* (/ d (sqrt l)) (fma (* (pow (/ (/ d D_m) M) -2.0) 0.25) t_0 1.0))
        (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (h / l) * -0.5;
	double t_1 = sqrt(-d);
	double tmp;
	if (h <= -3.5e+58) {
		tmp = fma(((M * (D_m * (0.5 / d))) * -0.5), (((D_m * 0.5) * (M * h)) / (l * d)), 1.0) * (sqrt((d / l)) * (t_1 / sqrt(-h)));
	} else if (h <= -1e-309) {
		tmp = ((sqrt((d / h)) * fma(t_0, pow(((d / D_m) * (2.0 / M)), -2.0), 1.0)) * t_1) / sqrt(-l);
	} else {
		tmp = ((d / sqrt(l)) * fma((pow(((d / D_m) / M), -2.0) * 0.25), t_0, 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(h / l) * -0.5)
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -3.5e+58)
		tmp = Float64(fma(Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5), Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0) * Float64(sqrt(Float64(d / l)) * Float64(t_1 / sqrt(Float64(-h)))));
	elseif (h <= -1e-309)
		tmp = Float64(Float64(Float64(sqrt(Float64(d / h)) * fma(t_0, (Float64(Float64(d / D_m) * Float64(2.0 / M)) ^ -2.0), 1.0)) * t_1) / sqrt(Float64(-l)));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) * fma(Float64((Float64(Float64(d / D_m) / M) ^ -2.0) * 0.25), t_0, 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -3.5e+58], N[(N[(N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-309], N[(N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[N[(N[(d / D$95$m), $MachinePrecision] * N[(2.0 / M), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision] * 0.25), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -3.4999999999999997e58

   1. Initial program 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 61.5

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

   8. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

   if -3.4999999999999997e58 < h < -1.000000000000002e-309

   1. Initial program 82.6%

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

   4. Applied rewrites 89.3%

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

   if -1.000000000000002e-309 < h

   1. Initial program 69.8%

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

   4. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell}}}}{\sqrt{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -3.5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{d}{D} \cdot \frac{2}{M}\right)}^{-2}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{d}{D\_m}}{M}\right)}^{-2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (* M (* D_m (/ 0.5 d))) -0.5)))
   (if (<= d -1.6e-129)
     (*
      (fma t_0 (/ (* (* D_m 0.5) (* M h)) (* l d)) 1.0)
      (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -4e-310)
       (*
        (fma t_0 (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0)
        (/ (- d) (sqrt (* l h))))
       (/
        (*
         (/ d (sqrt l))
         (fma (* (pow (/ (/ d D_m) M) -2.0) 0.25) (* (/ h l) -0.5) 1.0))
        (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M * (D_m * (0.5 / d))) * -0.5;
	double tmp;
	if (d <= -1.6e-129) {
		tmp = fma(t_0, (((D_m * 0.5) * (M * h)) / (l * d)), 1.0) * (sqrt((d / l)) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -4e-310) {
		tmp = fma(t_0, (((h / l) * M) * ((D_m / d) * 0.5)), 1.0) * (-d / sqrt((l * h)));
	} else {
		tmp = ((d / sqrt(l)) * fma((pow(((d / D_m) / M), -2.0) * 0.25), ((h / l) * -0.5), 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5)
	tmp = 0.0
	if (d <= -1.6e-129)
		tmp = Float64(fma(t_0, Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0) * Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -4e-310)
		tmp = Float64(fma(t_0, Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(l)) * fma(Float64((Float64(Float64(d / D_m) / M) ^ -2.0) * 0.25), Float64(Float64(h / l) * -0.5), 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[d, -1.6e-129], N[(N[(t$95$0 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(N[(t$95$0 * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[(N[(d / D$95$m), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6000000000000001e-129

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 80.4

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

   6. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.3

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

   8. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if -1.6000000000000001e-129 < d < -3.999999999999988e-310

   1. Initial program 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 52.6

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

   8. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-neg.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-unprod N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqr-neg N/A

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

      21. sqrt-unprod N/A

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

      22. rem-square-sqrt N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -3.999999999999988e-310 < d

   1. Initial program 69.8%

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

   4. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

   6. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.25 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell}}}}{\sqrt{h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}} \cdot \mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M}{\frac{d}{D\_m}} \cdot 0.5\\ t_1 := \left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(t\_0, t\_0 \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right) \cdot t\_2\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (/ M (/ d D_m)) 0.5))
        (t_1 (* (* M (* D_m (/ 0.5 d))) -0.5))
        (t_2 (sqrt (/ d l))))
   (if (<= d -1.6e-129)
     (*
      (fma t_1 (/ (* (* D_m 0.5) (* M h)) (* l d)) 1.0)
      (* t_2 (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -4e-310)
       (*
        (fma t_1 (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0)
        (/ (- d) (sqrt (* l h))))
       (/
        (* (* (fma t_0 (* t_0 (* (/ h l) -0.5)) 1.0) t_2) (sqrt d))
        (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M / (d / D_m)) * 0.5;
	double t_1 = (M * (D_m * (0.5 / d))) * -0.5;
	double t_2 = sqrt((d / l));
	double tmp;
	if (d <= -1.6e-129) {
		tmp = fma(t_1, (((D_m * 0.5) * (M * h)) / (l * d)), 1.0) * (t_2 * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -4e-310) {
		tmp = fma(t_1, (((h / l) * M) * ((D_m / d) * 0.5)), 1.0) * (-d / sqrt((l * h)));
	} else {
		tmp = ((fma(t_0, (t_0 * ((h / l) * -0.5)), 1.0) * t_2) * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M / Float64(d / D_m)) * 0.5)
	t_1 = Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5)
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.6e-129)
		tmp = Float64(fma(t_1, Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0) * Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -4e-310)
		tmp = Float64(fma(t_1, Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(Float64(fma(t_0, Float64(t_0 * Float64(Float64(h / l) * -0.5)), 1.0) * t_2) * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M / N[(d / D$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.6e-129], N[(N[(t$95$1 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(N[(t$95$1 * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6000000000000001e-129

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 80.4

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

   6. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.3

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

   8. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if -1.6000000000000001e-129 < d < -3.999999999999988e-310

   1. Initial program 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 52.6

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

   8. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-neg.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-unprod N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqr-neg N/A

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

      21. sqrt-unprod N/A

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

      22. rem-square-sqrt N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -3.999999999999988e-310 < d

   1. Initial program 69.8%

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

   4. Applied rewrites 79.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. inv-pow N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 80.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{M}{\frac{d}{D}} \cdot 0.5, \left(\frac{M}{\frac{d}{D}} \cdot 0.5\right) \cdot \left(\frac{h}{\ell} \cdot -0.5\right), 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5\\ t_1 := \mathsf{fma}\left(t\_0, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(t\_2 \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_2\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (* M (* D_m (/ 0.5 d))) -0.5))
        (t_1 (fma t_0 (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0))
        (t_2 (sqrt (/ d l))))
   (if (<= d -1.6e-129)
     (*
      (fma t_0 (/ (* (* D_m 0.5) (* M h)) (* l d)) 1.0)
      (* t_2 (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -4e-310)
       (* t_1 (/ (- d) (sqrt (* l h))))
       (* (* (/ (sqrt d) (sqrt h)) t_2) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M * (D_m * (0.5 / d))) * -0.5;
	double t_1 = fma(t_0, (((h / l) * M) * ((D_m / d) * 0.5)), 1.0);
	double t_2 = sqrt((d / l));
	double tmp;
	if (d <= -1.6e-129) {
		tmp = fma(t_0, (((D_m * 0.5) * (M * h)) / (l * d)), 1.0) * (t_2 * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -4e-310) {
		tmp = t_1 * (-d / sqrt((l * h)));
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * t_2) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5)
	t_1 = fma(t_0, Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0)
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.6e-129)
		tmp = Float64(fma(t_0, Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0) * Float64(t_2 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -4e-310)
		tmp = Float64(t_1 * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * t_2) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.6e-129], N[(N[(t$95$0 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(t$95$1 * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6000000000000001e-129

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 80.4

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

   6. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.3

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

   8. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if -1.6000000000000001e-129 < d < -3.999999999999988e-310

   1. Initial program 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 52.6

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

   8. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-neg.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-unprod N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqr-neg N/A

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

      21. sqrt-unprod N/A

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

      22. rem-square-sqrt N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -3.999999999999988e-310 < d

   1. Initial program 69.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 77.9

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

   8. Applied rewrites 77.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5\\ t_1 := \mathsf{fma}\left(t\_0, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right)\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (* M (* D_m (/ 0.5 d))) -0.5))
        (t_1 (fma t_0 (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0)))
   (if (<= d -1.6e-129)
     (*
      (fma t_0 (/ (* (* D_m 0.5) (* M h)) (* l d)) 1.0)
      (* (sqrt (/ d l)) (/ (sqrt (- d)) (sqrt (- h)))))
     (if (<= d -4e-310)
       (* t_1 (/ (- d) (sqrt (* l h))))
       (* (/ (/ d (sqrt h)) (sqrt l)) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M * (D_m * (0.5 / d))) * -0.5;
	double t_1 = fma(t_0, (((h / l) * M) * ((D_m / d) * 0.5)), 1.0);
	double tmp;
	if (d <= -1.6e-129) {
		tmp = fma(t_0, (((D_m * 0.5) * (M * h)) / (l * d)), 1.0) * (sqrt((d / l)) * (sqrt(-d) / sqrt(-h)));
	} else if (d <= -4e-310) {
		tmp = t_1 * (-d / sqrt((l * h)));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5)
	t_1 = fma(t_0, Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0)
	tmp = 0.0
	if (d <= -1.6e-129)
		tmp = Float64(fma(t_0, Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0) * Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h)))));
	elseif (d <= -4e-310)
		tmp = Float64(t_1 * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -1.6e-129], N[(N[(t$95$0 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(t$95$1 * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6000000000000001e-129

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 80.4

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

   6. Applied rewrites 80.4%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 80.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 82.3

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

   8. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 87.6

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

   10. Applied rewrites 87.6%

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

   if -1.6000000000000001e-129 < d < -3.999999999999988e-310

   1. Initial program 52.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 52.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 52.6

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

   8. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-div N/A

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

      11. sqrt-unprod N/A

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

      12. rem-square-sqrt N/A

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

      13. lift-sqrt.f64 N/A

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

      14. frac-2neg N/A

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

      15. lift-neg.f64 N/A

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

      16. rem-square-sqrt N/A

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

      17. sqrt-unprod N/A

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

      18. lift-neg.f64 N/A

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

      19. lift-neg.f64 N/A

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

      20. sqr-neg N/A

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

      21. sqrt-unprod N/A

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

      22. rem-square-sqrt N/A

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

      23. lower-/.f64 N/A

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

      24. lower-neg.f64 63.1

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

   10. Applied rewrites 63.1%

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

   if -3.999999999999988e-310 < d

   1. Initial program 69.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 70.5

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

   6. Applied rewrites 70.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 70.5

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 70.5

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

   8. Applied rewrites 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. pow1/2 N/A

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

      6. associate-*r/ N/A

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

      7. lift-sqrt.f64 N/A

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

      8. rem-exp-log N/A

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

      9. unpow1/2 N/A

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

      10. pow-prod-down N/A

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

      11. rem-exp-log N/A

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

      12. rem-square-sqrt N/A

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

      13. sqrt-unprod N/A

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

      14. sqr-neg N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. sqrt-unprod N/A

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

      18. rem-square-sqrt N/A

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

      19. lift-*.f64 N/A

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

      20. pow1/2 N/A

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

      21. lift-sqrt.f64 N/A

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

   10. Applied rewrites 76.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5\\ t_1 := \mathsf{fma}\left(t\_0, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right)\\ t_2 := \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(t\_0, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right)\\ t_3 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{-129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \frac{-d}{t\_3}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{d}{t\_3} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (* M (* D_m (/ 0.5 d))) -0.5))
        (t_1 (fma t_0 (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0))
        (t_2
         (*
          (* (sqrt (/ d h)) (sqrt (/ d l)))
          (fma t_0 (/ (* (* D_m 0.5) (* M h)) (* l d)) 1.0)))
        (t_3 (sqrt (* l h))))
   (if (<= d -3.6e-129)
     t_2
     (if (<= d -4e-310)
       (* t_1 (/ (- d) t_3))
       (if (<= d 3.2e-153) (* (/ d t_3) t_1) t_2)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M * (D_m * (0.5 / d))) * -0.5;
	double t_1 = fma(t_0, (((h / l) * M) * ((D_m / d) * 0.5)), 1.0);
	double t_2 = (sqrt((d / h)) * sqrt((d / l))) * fma(t_0, (((D_m * 0.5) * (M * h)) / (l * d)), 1.0);
	double t_3 = sqrt((l * h));
	double tmp;
	if (d <= -3.6e-129) {
		tmp = t_2;
	} else if (d <= -4e-310) {
		tmp = t_1 * (-d / t_3);
	} else if (d <= 3.2e-153) {
		tmp = (d / t_3) * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5)
	t_1 = fma(t_0, Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0)
	t_2 = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * fma(t_0, Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0))
	t_3 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -3.6e-129)
		tmp = t_2;
	elseif (d <= -4e-310)
		tmp = Float64(t_1 * Float64(Float64(-d) / t_3));
	elseif (d <= 3.2e-153)
		tmp = Float64(Float64(d / t_3) * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -3.6e-129], t$95$2, If[LessEqual[d, -4e-310], N[(t$95$1 * N[((-d) / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e-153], N[(N[(d / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.6e-129 or 3.1999999999999999e-153 < d

   1. Initial program 80.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 79.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 79.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 79.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 79.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 79.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 79.4

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 79.4%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\left(M \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right)}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\left(M \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right), 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(M \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right), 1\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{M \cdot h}{\ell}} \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right), 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{M \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}, 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{M \cdot h}{\ell} \cdot \left(\color{blue}{\frac{D}{d}} \cdot \frac{1}{2}\right), 1\right) \]

      8. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{M \cdot h}{\ell} \cdot \color{blue}{\frac{D \cdot \frac{1}{2}}{d}}, 1\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{\left(M \cdot h\right) \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}}, 1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{\left(M \cdot h\right) \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\color{blue}{\left(M \cdot h\right) \cdot \left(D \cdot \frac{1}{2}\right)}}{\ell \cdot d}, 1\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\color{blue}{\left(h \cdot M\right)} \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\color{blue}{\left(h \cdot M\right)} \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}, 1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\left(h \cdot M\right) \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}}{\ell \cdot d}, 1\right) \]

      15. lower-*.f64 82.3

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \frac{\left(h \cdot M\right) \cdot \left(D \cdot 0.5\right)}{\color{blue}{\ell \cdot d}}, 1\right) \]

   10. Applied rewrites 82.3%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{\left(h \cdot M\right) \cdot \left(D \cdot 0.5\right)}{\ell \cdot d}}, 1\right) \]

   if -3.6e-129 < d < -3.999999999999988e-310

   1. Initial program 52.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 52.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 52.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 52.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 52.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 52.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 52.6

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      10. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      14. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-d}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \left(-d\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      20. sqr-neg N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      21. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      22. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      23. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      24. lower-neg.f64 63.1

          \[\leadsto \frac{d}{\color{blue}{-\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   10. Applied rewrites 63.1%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   if -3.999999999999988e-310 < d < 3.1999999999999999e-153

   1. Initial program 44.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 47.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 47.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 47.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 47.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 47.8

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 47.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 47.8%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      10. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      14. lower-/.f64 61.9

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   10. Applied rewrites 61.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{-129}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5\\ t_1 := \mathsf{fma}\left(t\_0, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right)\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{-129}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(t\_0, \frac{\left(D\_m \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* (* M (* D_m (/ 0.5 d))) -0.5))
        (t_1 (fma t_0 (* (* (/ h l) M) (* (/ D_m d) 0.5)) 1.0)))
   (if (<= d -3.6e-129)
     (*
      (* (sqrt (/ d h)) (sqrt (/ d l)))
      (fma t_0 (/ (* (* D_m 0.5) (* M h)) (* l d)) 1.0))
     (if (<= d -4e-310)
       (* t_1 (/ (- d) (sqrt (* l h))))
       (* (/ (/ d (sqrt h)) (sqrt l)) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = (M * (D_m * (0.5 / d))) * -0.5;
	double t_1 = fma(t_0, (((h / l) * M) * ((D_m / d) * 0.5)), 1.0);
	double tmp;
	if (d <= -3.6e-129) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * fma(t_0, (((D_m * 0.5) * (M * h)) / (l * d)), 1.0);
	} else if (d <= -4e-310) {
		tmp = t_1 * (-d / sqrt((l * h)));
	} else {
		tmp = ((d / sqrt(h)) / sqrt(l)) * t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5)
	t_1 = fma(t_0, Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0)
	tmp = 0.0
	if (d <= -3.6e-129)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * fma(t_0, Float64(Float64(Float64(D_m * 0.5) * Float64(M * h)) / Float64(l * d)), 1.0));
	elseif (d <= -4e-310)
		tmp = Float64(t_1 * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(Float64(d / sqrt(h)) / sqrt(l)) * t_1);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[d, -3.6e-129], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(D$95$m * 0.5), $MachinePrecision] * N[(M * h), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(t$95$1 * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.6e-129

   1. Initial program 81.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 80.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 80.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 80.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 80.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 80.4

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 80.4

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 80.4%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right)}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\left(M \cdot \frac{h}{\ell}\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right)}, 1\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\left(M \cdot \frac{h}{\ell}\right)} \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right), 1\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(M \cdot \color{blue}{\frac{h}{\ell}}\right) \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right), 1\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{M \cdot h}{\ell}} \cdot \left(\frac{D}{d} \cdot \frac{1}{2}\right), 1\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{M \cdot h}{\ell} \cdot \color{blue}{\left(\frac{D}{d} \cdot \frac{1}{2}\right)}, 1\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{M \cdot h}{\ell} \cdot \left(\color{blue}{\frac{D}{d}} \cdot \frac{1}{2}\right), 1\right) \]

      8. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{M \cdot h}{\ell} \cdot \color{blue}{\frac{D \cdot \frac{1}{2}}{d}}, 1\right) \]

      9. frac-times N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{\left(M \cdot h\right) \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}}, 1\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{\left(M \cdot h\right) \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}}, 1\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\color{blue}{\left(M \cdot h\right) \cdot \left(D \cdot \frac{1}{2}\right)}}{\ell \cdot d}, 1\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\color{blue}{\left(h \cdot M\right)} \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}, 1\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\color{blue}{\left(h \cdot M\right)} \cdot \left(D \cdot \frac{1}{2}\right)}{\ell \cdot d}, 1\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \frac{\left(h \cdot M\right) \cdot \color{blue}{\left(D \cdot \frac{1}{2}\right)}}{\ell \cdot d}, 1\right) \]

      15. lower-*.f64 85.7

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \frac{\left(h \cdot M\right) \cdot \left(D \cdot 0.5\right)}{\color{blue}{\ell \cdot d}}, 1\right) \]

   10. Applied rewrites 85.7%

       \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \color{blue}{\frac{\left(h \cdot M\right) \cdot \left(D \cdot 0.5\right)}{\ell \cdot d}}, 1\right) \]

   if -3.6e-129 < d < -3.999999999999988e-310

   1. Initial program 52.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 52.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 52.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 52.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 52.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 52.6

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 52.6

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 52.6%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      10. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      14. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-d}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \left(-d\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      20. sqr-neg N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      21. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      22. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      23. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      24. lower-neg.f64 63.1

          \[\leadsto \frac{d}{\color{blue}{-\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   10. Applied rewrites 63.1%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   if -3.999999999999988e-310 < d

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 70.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 70.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 70.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 70.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 70.5

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 70.5

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 70.5%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{h}} \cdot {d}^{\frac{1}{2}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}} \cdot {d}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      8. rem-exp-log N/A

         \[\leadsto \frac{\sqrt{\color{blue}{e^{\log \left(\frac{d}{h}\right)}}} \cdot {d}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      9. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)}\right)}^{\frac{1}{2}}} \cdot {d}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\color{blue}{{\left(e^{\log \left(\frac{d}{h}\right)} \cdot d\right)}^{\frac{1}{2}}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      11. rem-exp-log N/A

          \[\leadsto \frac{{\left(\color{blue}{\frac{d}{h}} \cdot d\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \color{blue}{\left(\sqrt{d} \cdot \sqrt{d}\right)}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      13. sqrt-unprod N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \color{blue}{\sqrt{d \cdot d}}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      14. sqr-neg N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right) \cdot \left(\mathsf{neg}\left(d\right)\right)}}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \sqrt{\color{blue}{\left(-d\right)} \cdot \left(\mathsf{neg}\left(d\right)\right)}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      16. lift-neg.f64 N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \sqrt{\left(-d\right) \cdot \color{blue}{\left(-d\right)}}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \color{blue}{\left(\sqrt{-d} \cdot \sqrt{-d}\right)}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      18. rem-square-sqrt N/A

          \[\leadsto \frac{{\left(\frac{d}{h} \cdot \color{blue}{\left(-d\right)}\right)}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{{\color{blue}{\left(\frac{d}{h} \cdot \left(-d\right)\right)}}^{\frac{1}{2}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      20. pow1/2 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      21. lift-sqrt.f64 N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   10. Applied rewrites 76.3%

       \[\leadsto \color{blue}{\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{-129}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \frac{\left(D \cdot 0.5\right) \cdot \left(M \cdot h\right)}{\ell \cdot d}, 1\right)\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(M \cdot \left(D\_m \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D\_m}{d} \cdot 0.5\right), 1\right)\\ t_1 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t\_0 \cdot \frac{-d}{t\_1}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{d}{t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* (* M (* D_m (/ 0.5 d))) -0.5)
          (* (* (/ h l) M) (* (/ D_m d) 0.5))
          1.0))
        (t_1 (sqrt (* l h))))
   (if (<= l -5e-310)
     (* t_0 (/ (- d) t_1))
     (if (<= l 1.1e+203)
       (* (/ d t_1) t_0)
       (* (sqrt (/ d h)) (sqrt (/ d l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = fma(((M * (D_m * (0.5 / d))) * -0.5), (((h / l) * M) * ((D_m / d) * 0.5)), 1.0);
	double t_1 = sqrt((l * h));
	double tmp;
	if (l <= -5e-310) {
		tmp = t_0 * (-d / t_1);
	} else if (l <= 1.1e+203) {
		tmp = (d / t_1) * t_0;
	} else {
		tmp = sqrt((d / h)) * sqrt((d / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = fma(Float64(Float64(M * Float64(D_m * Float64(0.5 / d))) * -0.5), Float64(Float64(Float64(h / l) * M) * Float64(Float64(D_m / d) * 0.5)), 1.0)
	t_1 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(t_0 * Float64(Float64(-d) / t_1));
	elseif (l <= 1.1e+203)
		tmp = Float64(Float64(d / t_1) * t_0);
	else
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(M * N[(D$95$m * N[(0.5 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(N[(h / l), $MachinePrecision] * M), $MachinePrecision] * N[(N[(D$95$m / d), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(t$95$0 * N[((-d) / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.1e+203], N[(N[(d / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 72.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 72.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 72.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 72.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 72.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 72.0

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 72.0

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 72.0%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      10. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      14. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-d}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      16. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{\sqrt{-d} \cdot \sqrt{-d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      17. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \left(-d\right)}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      18. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \left(-d\right)}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      20. sqr-neg N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      21. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      22. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      23. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{d}{\mathsf{neg}\left(\sqrt{\ell \cdot h}\right)}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      24. lower-neg.f64 72.5

          \[\leadsto \frac{d}{\color{blue}{-\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   10. Applied rewrites 72.5%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   if -4.999999999999985e-310 < l < 1.10000000000000002e203

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right)\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)}\right)\right) + 1\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \frac{h}{\ell}\right) + 1\right) \]

      9. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) + 1\right) \]

      10. associate-*l* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right)} + 1\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)} + 1\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{M \cdot D}{2 \cdot d}, \frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}, 1\right)} \]

   4. Applied rewrites 68.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 68.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-sqrt.f64 68.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   6. Applied rewrites 68.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. metadata-eval 68.3

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lower-sqrt.f64 68.3

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   8. Applied rewrites 68.3%

      \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      7. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      10. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      11. sqrt-unprod N/A

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{d}}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(\frac{-1}{2} \cdot \left(\left(\frac{\frac{1}{2}}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot \frac{1}{2}\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

      14. lower-/.f64 70.6

          \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   10. Applied rewrites 70.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \cdot \mathsf{fma}\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right), \left(\frac{D}{d} \cdot 0.5\right) \cdot \left(M \cdot \frac{h}{\ell}\right), 1\right) \]

   if 1.10000000000000002e203 < l

   1. Initial program 79.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 24.7

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 24.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.9%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+203}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right) \cdot -0.5, \left(\frac{h}{\ell} \cdot M\right) \cdot \left(\frac{D}{d} \cdot 0.5\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(-0.125 \cdot M\right) \cdot \left(\left(\left(\frac{D\_m}{d} \cdot D\_m\right) \cdot \sqrt{\ell \cdot h}\right) \cdot M\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.06e-206)
   (/ (/ (- d) (sqrt (- h))) (sqrt (- l)))
   (if (<= d -4e-310)
     (/ d (sqrt (sqrt (* (* l h) (* l h)))))
     (if (<= d 1.75e+14)
       (/ (* (* -0.125 M) (* (* (* (/ D_m d) D_m) (sqrt (* l h))) M)) (* l l))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	} else if (d <= -4e-310) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else if (d <= 1.75e+14) {
		tmp = ((-0.125 * M) * ((((D_m / d) * D_m) * sqrt((l * h))) * M)) / (l * l);
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.06d-206)) then
        tmp = (-d / sqrt(-h)) / sqrt(-l)
    else if (d <= (-4d-310)) then
        tmp = d / sqrt(sqrt(((l * h) * (l * h))))
    else if (d <= 1.75d+14) then
        tmp = (((-0.125d0) * m) * ((((d_m / d) * d_m) * sqrt((l * h))) * m)) / (l * l)
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = (-d / Math.sqrt(-h)) / Math.sqrt(-l);
	} else if (d <= -4e-310) {
		tmp = d / Math.sqrt(Math.sqrt(((l * h) * (l * h))));
	} else if (d <= 1.75e+14) {
		tmp = ((-0.125 * M) * ((((D_m / d) * D_m) * Math.sqrt((l * h))) * M)) / (l * l);
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.06e-206:
		tmp = (-d / math.sqrt(-h)) / math.sqrt(-l)
	elif d <= -4e-310:
		tmp = d / math.sqrt(math.sqrt(((l * h) * (l * h))))
	elif d <= 1.75e+14:
		tmp = ((-0.125 * M) * ((((D_m / d) * D_m) * math.sqrt((l * h))) * M)) / (l * l)
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.06e-206)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-h))) / sqrt(Float64(-l)));
	elseif (d <= -4e-310)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	elseif (d <= 1.75e+14)
		tmp = Float64(Float64(Float64(-0.125 * M) * Float64(Float64(Float64(Float64(D_m / d) * D_m) * sqrt(Float64(l * h))) * M)) / Float64(l * l));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.06e-206)
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	elseif (d <= -4e-310)
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	elseif (d <= 1.75e+14)
		tmp = ((-0.125 * M) * ((((D_m / d) * D_m) * sqrt((l * h))) * M)) / (l * l);
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.06e-206], N[(N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e+14], N[(N[(N[(-0.125 * M), $MachinePrecision] * N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * D$95$m), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.06e-206

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 9.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\color{blue}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.7%

      \[\leadsto \frac{\frac{d}{-\sqrt{-h}}}{\sqrt{\color{blue}{-\ell}}} \]

   if -1.06e-206 < d < -3.999999999999988e-310

   1. Initial program 46.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 25.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}} \]

   if -3.999999999999988e-310 < d < 1.75e14

   1. Initial program 59.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]

   5. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\color{blue}{\ell} \cdot \ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.1%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{d} \cdot \sqrt{\ell \cdot h}\right)}{\color{blue}{\ell} \cdot \ell} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.5%

       \[\leadsto \frac{\left(\left(\left(\frac{D}{d} \cdot D\right) \cdot \sqrt{\ell \cdot h}\right) \cdot M\right) \cdot \left(-0.125 \cdot M\right)}{\ell \cdot \ell} \]

   if 1.75e14 < d

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 65.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.2%

      \[\leadsto \frac{\frac{d}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(-0.125 \cdot M\right) \cdot \left(\left(\left(\frac{D}{d} \cdot D\right) \cdot \sqrt{\ell \cdot h}\right) \cdot M\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-291}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(\frac{D\_m \cdot D\_m}{d} \cdot \sqrt{\ell \cdot h}\right) \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.06e-206)
   (/ (/ (- d) (sqrt (- h))) (sqrt (- l)))
   (if (<= d 9.2e-291)
     (/ d (sqrt (sqrt (* (* l h) (* l h)))))
     (if (<= d 1.75e+14)
       (/ (* (* (/ (* D_m D_m) d) (sqrt (* l h))) (* -0.125 (* M M))) (* l l))
       (/ (/ d (sqrt l)) (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	} else if (d <= 9.2e-291) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else if (d <= 1.75e+14) {
		tmp = ((((D_m * D_m) / d) * sqrt((l * h))) * (-0.125 * (M * M))) / (l * l);
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.06d-206)) then
        tmp = (-d / sqrt(-h)) / sqrt(-l)
    else if (d <= 9.2d-291) then
        tmp = d / sqrt(sqrt(((l * h) * (l * h))))
    else if (d <= 1.75d+14) then
        tmp = ((((d_m * d_m) / d) * sqrt((l * h))) * ((-0.125d0) * (m * m))) / (l * l)
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = (-d / Math.sqrt(-h)) / Math.sqrt(-l);
	} else if (d <= 9.2e-291) {
		tmp = d / Math.sqrt(Math.sqrt(((l * h) * (l * h))));
	} else if (d <= 1.75e+14) {
		tmp = ((((D_m * D_m) / d) * Math.sqrt((l * h))) * (-0.125 * (M * M))) / (l * l);
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.06e-206:
		tmp = (-d / math.sqrt(-h)) / math.sqrt(-l)
	elif d <= 9.2e-291:
		tmp = d / math.sqrt(math.sqrt(((l * h) * (l * h))))
	elif d <= 1.75e+14:
		tmp = ((((D_m * D_m) / d) * math.sqrt((l * h))) * (-0.125 * (M * M))) / (l * l)
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.06e-206)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-h))) / sqrt(Float64(-l)));
	elseif (d <= 9.2e-291)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	elseif (d <= 1.75e+14)
		tmp = Float64(Float64(Float64(Float64(Float64(D_m * D_m) / d) * sqrt(Float64(l * h))) * Float64(-0.125 * Float64(M * M))) / Float64(l * l));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.06e-206)
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	elseif (d <= 9.2e-291)
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	elseif (d <= 1.75e+14)
		tmp = ((((D_m * D_m) / d) * sqrt((l * h))) * (-0.125 * (M * M))) / (l * l);
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.06e-206], N[(N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-291], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e+14], N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] * N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.06e-206

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 9.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\color{blue}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.7%

      \[\leadsto \frac{\frac{d}{-\sqrt{-h}}}{\sqrt{\color{blue}{-\ell}}} \]

   if -1.06e-206 < d < 9.2000000000000003e-291

   1. Initial program 47.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 21.6

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 25.3%

      \[\leadsto \frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}} \]

   if 9.2000000000000003e-291 < d < 1.75e14

   1. Initial program 60.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]

   5. Applied rewrites 38.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \frac{D \cdot D}{d}, \sqrt{\ell \cdot h}, \sqrt{\frac{{\ell}^{3}}{h}} \cdot d\right)}{\ell \cdot \ell}} \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right)}{\color{blue}{\ell} \cdot \ell} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.6%

      \[\leadsto \frac{\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(\frac{D \cdot D}{d} \cdot \sqrt{\ell \cdot h}\right)}{\color{blue}{\ell} \cdot \ell} \]

   if 1.75e14 < d

   1. Initial program 79.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 65.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.2%

      \[\leadsto \frac{\frac{d}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-291}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(\frac{D \cdot D}{d} \cdot \sqrt{\ell \cdot h}\right) \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 48.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.06e-206)
   (/ (/ (- d) (sqrt (- h))) (sqrt (- l)))
   (if (<= d -4e-310)
     (/ d (sqrt (sqrt (* (* l h) (* l h)))))
     (/ (/ d (sqrt l)) (sqrt h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	} else if (d <= -4e-310) {
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.06d-206)) then
        tmp = (-d / sqrt(-h)) / sqrt(-l)
    else if (d <= (-4d-310)) then
        tmp = d / sqrt(sqrt(((l * h) * (l * h))))
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = (-d / Math.sqrt(-h)) / Math.sqrt(-l);
	} else if (d <= -4e-310) {
		tmp = d / Math.sqrt(Math.sqrt(((l * h) * (l * h))));
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.06e-206:
		tmp = (-d / math.sqrt(-h)) / math.sqrt(-l)
	elif d <= -4e-310:
		tmp = d / math.sqrt(math.sqrt(((l * h) * (l * h))))
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.06e-206)
		tmp = Float64(Float64(Float64(-d) / sqrt(Float64(-h))) / sqrt(Float64(-l)));
	elseif (d <= -4e-310)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(l * h) * Float64(l * h)))));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.06e-206)
		tmp = (-d / sqrt(-h)) / sqrt(-l);
	elseif (d <= -4e-310)
		tmp = d / sqrt(sqrt(((l * h) * (l * h))));
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.06e-206], N[(N[((-d) / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4e-310], N[(d / N[Sqrt[N[Sqrt[N[(N[(l * h), $MachinePrecision] * N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.06e-206

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 9.0

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 9.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto \frac{\sqrt{\frac{d}{h} \cdot \left(-d\right)}}{\color{blue}{\sqrt{-\ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.7%

      \[\leadsto \frac{\frac{d}{-\sqrt{-h}}}{\sqrt{\color{blue}{-\ell}}} \]

   if -1.06e-206 < d < -3.999999999999988e-310

   1. Initial program 46.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 25.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.5%

      \[\leadsto \frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}} \]

   if -3.999999999999988e-310 < d

   1. Initial program 69.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 42.1

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.8%

      \[\leadsto \frac{\frac{d}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{-d}{\sqrt{-h}}}{\sqrt{-\ell}}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(\ell \cdot h\right) \cdot \left(\ell \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d 4.2e-179)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ (/ d (sqrt l)) (sqrt h))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 4.2e-179) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = (d / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 4.2d-179) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = (d / sqrt(l)) / sqrt(h)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 4.2e-179) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = (d / Math.sqrt(l)) / Math.sqrt(h);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= 4.2e-179:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = (d / math.sqrt(l)) / math.sqrt(h)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= 4.2e-179)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(Float64(d / sqrt(l)) / sqrt(h));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= 4.2e-179)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = (d / sqrt(l)) / sqrt(h);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, 4.2e-179], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.1999999999999997e-179

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 43.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.1999999999999997e-179 < d

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 52.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.9%

      \[\leadsto \frac{\frac{d}{\sqrt{\ell}}}{\color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 45.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d 4.2e-179)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (/ d (* (sqrt h) (sqrt l)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 4.2e-179) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 4.2d-179) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 4.2e-179) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= 4.2e-179:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= 4.2e-179)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= 4.2e-179)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, 4.2e-179], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 4.1999999999999997e-179

   1. Initial program 67.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 43.4

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if 4.1999999999999997e-179 < d

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 52.5

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 52.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 62.8%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 4.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 41.5% accurate, 10.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d -1.06e-206) (* (sqrt (/ 1.0 (* l h))) (- d)) (/ d (sqrt (* l h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= (-1.06d-206)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= -1.06e-206) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= -1.06e-206:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= -1.06e-206)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= -1.06e-206)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, -1.06e-206], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.06e-206

   1. Initial program 77.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-d\right) \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      9. *-commutative N/A

         \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      10. lower-*.f64 56.1

          \[\leadsto \left(-d\right) \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(-d\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}} \]

   if -1.06e-206 < d

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      6. lower-*.f64 39.4

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.06 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 25.7% accurate, 15.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m) :precision binary64 (/ d (sqrt (* l h))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	return d / sqrt((l * h));
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_m
    code = d / sqrt((l * h))
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	return d / Math.sqrt((l * h));
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	return d / math.sqrt((l * h))

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp = code(d, h, l, M, D_m)
	tmp = d / sqrt((l * h));
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.3%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   6. lower-*.f64 25.5

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 25.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 25.8%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(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)))))