Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 80.1%

Time: 17.2s

Alternatives: 23

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.8% 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: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell}\\ \mathbf{if}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - t\_0 \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - t\_0 \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ (* (* (/ D d) M) 0.5) l)))
   (if (<= d -5e-311)
     (*
      (* (pow (/ d h) (pow 2.0 -1.0)) (* (sqrt (- d)) (sqrt (/ -1.0 l))))
      (- 1.0 (* t_0 (/ (* (* M (* 0.25 D)) h) d))))
     (if (<= d 7.6e-44)
       (/
        (*
         (/
          (*
           (fma (* 0.25 (pow (/ d (* M D)) -2.0)) (* (/ h l) -0.5) 1.0)
           (sqrt d))
          (sqrt h))
         (sqrt d))
        (sqrt l))
       (*
        (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0)))
        (- 1.0 (* t_0 (/ (* (* 0.5 (* D 0.5)) (/ M d)) (pow h -1.0)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (((D / d) * M) * 0.5) / l;
	double tmp;
	if (d <= -5e-311) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (sqrt(-d) * sqrt((-1.0 / l)))) * (1.0 - (t_0 * (((M * (0.25 * D)) * h) / d)));
	} else if (d <= 7.6e-44) {
		tmp = (((fma((0.25 * pow((d / (M * D)), -2.0)), ((h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - (t_0 * (((0.5 * (D * 0.5)) * (M / d)) / pow(h, -1.0))));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(Float64(D / d) * M) * 0.5) / l)
	tmp = 0.0
	if (d <= -5e-311)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(sqrt(Float64(-d)) * sqrt(Float64(-1.0 / l)))) * Float64(1.0 - Float64(t_0 * Float64(Float64(Float64(M * Float64(0.25 * D)) * h) / d))));
	elseif (d <= 7.6e-44)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(0.25 * (Float64(d / Float64(M * D)) ^ -2.0)), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(t_0 * Float64(Float64(Float64(0.5 * Float64(D * 0.5)) * Float64(M / d)) / (h ^ -1.0)))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[d, -5e-311], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(N[(M * N[(0.25 * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.6e-44], N[(N[(N[(N[(N[(N[(0.25 * N[Power[N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(N[(N[(0.5 * N[(D * 0.5), $MachinePrecision]), $MachinePrecision] * N[(M / d), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000023e-311

   1. Initial program 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 69.4

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 69.4

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

   6. Applied rewrites 69.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 69.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 69.4

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

   8. Applied rewrites 69.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. distribute-neg-frac2 N/A

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

      6. unpow-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 77.2

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

      14. lift-neg.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow-1 N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 77.2

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

   10. Applied rewrites 77.2%

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

   if -5.00000000000023e-311 < d < 7.6000000000000002e-44

   1. Initial program 60.7%

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

   4. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 81.8%

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

   if 7.6000000000000002e-44 < d

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 94.0

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

   6. Applied rewrites 94.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right)\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(0.5 \cdot \left(D \cdot 0.5\right)\right) \cdot \frac{M}{d}}{{h}^{-1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{{h}^{-1}}{\ell}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (- 1.0 (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      5e-228)
   (* (sqrt (/ (pow h -1.0) l)) d)
   (* (sqrt (/ d l)) (sqrt (/ d h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e-228) {
		tmp = sqrt((pow(h, -1.0) / l)) * d;
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 5d-228) then
        tmp = sqrt(((h ** (-1.0d0)) / l)) * d
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e-228) {
		tmp = Math.sqrt((Math.pow(h, -1.0) / l)) * d;
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e-228:
		tmp = math.sqrt((math.pow(h, -1.0) / l)) * d
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e-228)
		tmp = Float64(sqrt(Float64((h ^ -1.0) / l)) * d);
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 5e-228)
		tmp = sqrt(((h ^ -1.0) / l)) * d;
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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], 5e-228], N[(N[Sqrt[N[(N[Power[h, -1.0], $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < 4.99999999999999972e-228

   1. Initial program 86.6%

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

   3. 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 19.5

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

   5. Applied rewrites 19.5%

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

   7. Applied rewrites 19.5%

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

   if 4.99999999999999972e-228 < (*.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 56.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 33.3

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

   5. Applied rewrites 33.3%

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

   7. Applied rewrites 61.8%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{\frac{{h}^{-1}}{\ell}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{D}{\ell}}{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (- 1.0 (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-249)
   (/ (/ (* (* (* (* (sqrt h) -0.125) M) (* M D)) (/ D l)) d) (sqrt l))
   (* (sqrt (/ d l)) (sqrt (/ d h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249) {
		tmp = (((((sqrt(h) * -0.125) * M) * (M * D)) * (D / l)) / d) / sqrt(l);
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-249)) then
        tmp = (((((sqrt(h) * (-0.125d0)) * m) * (m * d_1)) * (d_1 / l)) / d) / sqrt(l)
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249) {
		tmp = (((((Math.sqrt(h) * -0.125) * M) * (M * D)) * (D / l)) / d) / Math.sqrt(l);
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249:
		tmp = (((((math.sqrt(h) * -0.125) * M) * (M * D)) * (D / l)) / d) / math.sqrt(l)
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-249)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(sqrt(h) * -0.125) * M) * Float64(M * D)) * Float64(D / l)) / d) / sqrt(l));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-249)
		tmp = (((((sqrt(h) * -0.125) * M) * (M * D)) * (D / l)) / d) / sqrt(l);
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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], -5e-249], N[(N[(N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * N[(D / l), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -4.9999999999999999e-249

   1. Initial program 93.6%

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 39.1%

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

   11. Applied rewrites 44.2%

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

   if -4.9999999999999999e-249 < (*.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 56.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 60.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot \frac{D}{\ell}}{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot D}{\ell \cdot d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (- 1.0 (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-249)
   (/ (/ (* (* (* (* (sqrt h) -0.125) M) (* M D)) D) (* l d)) (sqrt l))
   (* (sqrt (/ d l)) (sqrt (/ d h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249) {
		tmp = (((((sqrt(h) * -0.125) * M) * (M * D)) * D) / (l * d)) / sqrt(l);
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-249)) then
        tmp = (((((sqrt(h) * (-0.125d0)) * m) * (m * d_1)) * d_1) / (l * d)) / sqrt(l)
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249) {
		tmp = (((((Math.sqrt(h) * -0.125) * M) * (M * D)) * D) / (l * d)) / Math.sqrt(l);
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249:
		tmp = (((((math.sqrt(h) * -0.125) * M) * (M * D)) * D) / (l * d)) / math.sqrt(l)
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-249)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(sqrt(h) * -0.125) * M) * Float64(M * D)) * D) / Float64(l * d)) / sqrt(l));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-249)
		tmp = (((((sqrt(h) * -0.125) * M) * (M * D)) * D) / (l * d)) / sqrt(l);
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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], -5e-249], N[(N[(N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -4.9999999999999999e-249

   1. Initial program 93.6%

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 39.1%

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

   11. Applied rewrites 44.0%

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

   if -4.9999999999999999e-249 < (*.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 56.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 60.5%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot D}{\ell \cdot d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\frac{\left(-0.125 \cdot \sqrt{h}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell \cdot d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (- 1.0 (* (* (pow 2.0 -1.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l))))
      -5e-249)
   (/ (* (* -0.125 (sqrt h)) (* (* (* M M) D) (/ D (* l d)))) (sqrt l))
   (* (sqrt (/ d l)) (sqrt (/ d h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249) {
		tmp = ((-0.125 * sqrt(h)) * (((M * M) * D) * (D / (l * d)))) / sqrt(l);
	} else {
		tmp = sqrt((d / l)) * sqrt((d / h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-249)) then
        tmp = (((-0.125d0) * sqrt(h)) * (((m * m) * d_1) * (d_1 / (l * d)))) / sqrt(l)
    else
        tmp = sqrt((d / l)) * sqrt((d / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249) {
		tmp = ((-0.125 * Math.sqrt(h)) * (((M * M) * D) * (D / (l * d)))) / Math.sqrt(l);
	} else {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-249:
		tmp = ((-0.125 * math.sqrt(h)) * (((M * M) * D) * (D / (l * d)))) / math.sqrt(l)
	else:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-249)
		tmp = Float64(Float64(Float64(-0.125 * sqrt(h)) * Float64(Float64(Float64(M * M) * D) * Float64(D / Float64(l * d)))) / sqrt(l));
	else
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-249)
		tmp = ((-0.125 * sqrt(h)) * (((M * M) * D) * (D / (l * d)))) / sqrt(l);
	else
		tmp = sqrt((d / l)) * sqrt((d / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $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], -5e-249], N[(N[(N[(-0.125 * N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * D), $MachinePrecision] * N[(D / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $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)))) < -4.9999999999999999e-249

   1. Initial program 93.6%

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 41.1

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

   7. Applied rewrites 41.1%

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

   9. Applied rewrites 39.1%

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

   if -4.9999999999999999e-249 < (*.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 56.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 60.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-249}:\\ \;\;\;\;\frac{\left(-0.125 \cdot \sqrt{h}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{\ell \cdot d}\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -8.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \left(t\_0 \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -8.5e+128)
     (/
      (*
       (*
        (fma (* -0.5 (/ h l)) (pow (* (/ d M) (/ 2.0 D)) -2.0) 1.0)
        (sqrt (/ d l)))
       t_0)
      (sqrt (- h)))
     (if (<= h -5e-310)
       (*
        (* (pow (/ d h) (pow 2.0 -1.0)) (* t_0 (sqrt (/ -1.0 l))))
        (- 1.0 (* (/ (* (* (/ D d) M) 0.5) l) (/ (* (* M (* 0.25 D)) h) d))))
       (/
        (*
         (/
          (*
           (fma (* 0.25 (pow (/ d (* M D)) -2.0)) (* (/ h l) -0.5) 1.0)
           (sqrt d))
          (sqrt h))
         (sqrt d))
        (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -8.5e+128) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M) * (2.0 / D)), -2.0), 1.0) * sqrt((d / l))) * t_0) / sqrt(-h);
	} else if (h <= -5e-310) {
		tmp = (pow((d / h), pow(2.0, -1.0)) * (t_0 * sqrt((-1.0 / l)))) * (1.0 - (((((D / d) * M) * 0.5) / l) * (((M * (0.25 * D)) * h) / d)));
	} else {
		tmp = (((fma((0.25 * pow((d / (M * D)), -2.0)), ((h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -8.5e+128)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * t_0) / sqrt(Float64(-h)));
	elseif (h <= -5e-310)
		tmp = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * Float64(t_0 * sqrt(Float64(-1.0 / l)))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D / d) * M) * 0.5) / l) * Float64(Float64(Float64(M * Float64(0.25 * D)) * h) / d))));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(0.25 * (Float64(d / Float64(M * D)) ^ -2.0)), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -8.5e+128], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -5e-310], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(-1.0 / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M * N[(0.25 * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.25 * N[Power[N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -8.50000000000000045e128

   1. Initial program 58.3%

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

   4. Applied rewrites 71.9%

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

   if -8.50000000000000045e128 < h < -4.999999999999985e-310

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.1

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 75.1

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

   6. Applied rewrites 75.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 75.1

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 75.1

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

   8. Applied rewrites 75.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. distribute-neg-frac2 N/A

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

      6. unpow-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-prod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 83.3

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

      14. lift-neg.f64 N/A

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

      15. neg-mul-1 N/A

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

      16. lift-pow.f64 N/A

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

      17. unpow-1 N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 83.3

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

   10. Applied rewrites 83.3%

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

   if -4.999999999999985e-310 < h

   1. Initial program 72.1%

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 83.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -8.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \left(\sqrt{-d} \cdot \sqrt{\frac{-1}{\ell}}\right)\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -5e-311)
   (/
    (*
     (*
      (fma (* -0.5 (/ h l)) (pow (* (/ d M) (/ 2.0 D)) -2.0) 1.0)
      (sqrt (/ d l)))
     (sqrt (- d)))
    (sqrt (- h)))
   (if (<= d 7.6e-44)
     (/
      (*
       (/
        (*
         (fma (* 0.25 (pow (/ d (* M D)) -2.0)) (* (/ h l) -0.5) 1.0)
         (sqrt d))
        (sqrt h))
       (sqrt d))
      (sqrt l))
     (*
      (* (/ (sqrt d) (sqrt h)) (pow (/ d l) (pow 2.0 -1.0)))
      (- 1.0 (* (/ (* (* (/ D d) M) 0.5) l) (/ (* (* M (* 0.25 D)) h) d)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -5e-311) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M) * (2.0 / D)), -2.0), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else if (d <= 7.6e-44) {
		tmp = (((fma((0.25 * pow((d / (M * D)), -2.0)), ((h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l);
	} else {
		tmp = ((sqrt(d) / sqrt(h)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - (((((D / d) * M) * 0.5) / l) * (((M * (0.25 * D)) * h) / d)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -5e-311)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 7.6e-44)
		tmp = Float64(Float64(Float64(Float64(fma(Float64(0.25 * (Float64(d / Float64(M * D)) ^ -2.0)), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l));
	else
		tmp = Float64(Float64(Float64(sqrt(d) / sqrt(h)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D / d) * M) * 0.5) / l) * Float64(Float64(Float64(M * Float64(0.25 * D)) * h) / d))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -5e-311], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.6e-44], N[(N[(N[(N[(N[(N[(0.25 * N[Power[N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D / d), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M * N[(0.25 * D), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.00000000000023e-311

   1. Initial program 66.8%

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

   4. Applied rewrites 74.7%

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

   if -5.00000000000023e-311 < d < 7.6000000000000002e-44

   1. Initial program 60.7%

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

   4. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 81.8%

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

   if 7.6000000000000002e-44 < d

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 85.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.7

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

   6. Applied rewrites 85.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/2 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 91.7

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

   8. Applied rewrites 91.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{d}}{\sqrt{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{d}}{\sqrt{h}} \cdot \sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6e-305)
   (/
    (*
     (*
      (fma (* -0.5 (/ h l)) (pow (* (/ d M) (/ 2.0 D)) -2.0) 1.0)
      (sqrt (/ d l)))
     (sqrt (- d)))
    (sqrt (- h)))
   (/
    (*
     (/
      (* (fma (* 0.25 (pow (/ d (* M D)) -2.0)) (* (/ h l) -0.5) 1.0) (sqrt d))
      (sqrt h))
     (sqrt d))
    (sqrt l))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e-305) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M) * (2.0 / D)), -2.0), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else {
		tmp = (((fma((0.25 * pow((d / (M * D)), -2.0)), ((h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6e-305)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(0.25 * (Float64(d / Float64(M * D)) ^ -2.0)), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(d)) / sqrt(h)) * sqrt(d)) / sqrt(l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6e-305], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.25 * N[Power[N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.0000000000000002e-305

   1. Initial program 67.4%

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

   4. Applied rewrites 75.3%

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

   if -6.0000000000000002e-305 < l

   1. Initial program 71.5%

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

   4. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 83.0%

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

Alternative 9: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot 0.25, {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}, 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6e-305)
   (/
    (*
     (*
      (fma (* -0.5 (/ h l)) (pow (* (/ d M) (/ 2.0 D)) -2.0) 1.0)
      (sqrt (/ d l)))
     (sqrt (- d)))
    (sqrt (- h)))
   (/
    (/
     (* (fma (* (* (/ h l) -0.5) 0.25) (pow (/ (/ d M) D) -2.0) 1.0) d)
     (sqrt l))
    (sqrt h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e-305) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M) * (2.0 / D)), -2.0), 1.0) * sqrt((d / l))) * sqrt(-d)) / sqrt(-h);
	} else {
		tmp = ((fma((((h / l) * -0.5) * 0.25), pow(((d / M) / D), -2.0), 1.0) * d) / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6e-305)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.5) * 0.25), (Float64(Float64(d / M) / D) ^ -2.0), 1.0) * d) / sqrt(l)) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6e-305], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * 0.25), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.0000000000000002e-305

   1. Initial program 67.4%

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

   4. Applied rewrites 75.3%

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

   if -6.0000000000000002e-305 < l

   1. Initial program 71.5%

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

   4. Applied rewrites 73.2%

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

   6. Applied rewrites 68.5%

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

   8. Applied rewrites 80.4%

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

Alternative 10: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-305}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{-d}}{\sqrt{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot 0.25, {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}, 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6e-305)
   (/
    (*
     (*
      (fma (* -0.5 (/ h l)) (pow (* (/ d M) (/ 2.0 D)) -2.0) 1.0)
      (sqrt (/ d h)))
     (sqrt (- d)))
    (sqrt (- l)))
   (/
    (/
     (* (fma (* (* (/ h l) -0.5) 0.25) (pow (/ (/ d M) D) -2.0) 1.0) d)
     (sqrt l))
    (sqrt h))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e-305) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M) * (2.0 / D)), -2.0), 1.0) * sqrt((d / h))) * sqrt(-d)) / sqrt(-l);
	} else {
		tmp = ((fma((((h / l) * -0.5) * 0.25), pow(((d / M) / D), -2.0), 1.0) * d) / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6e-305)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(-d))) / sqrt(Float64(-l)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.5) * 0.25), (Float64(Float64(d / M) / D) ^ -2.0), 1.0) * d) / sqrt(l)) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6e-305], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * 0.25), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.0000000000000002e-305

   1. Initial program 67.4%

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

   4. Applied rewrites 73.0%

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

   if -6.0000000000000002e-305 < l

   1. Initial program 71.5%

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

   4. Applied rewrites 73.2%

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

   6. Applied rewrites 68.5%

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

   8. Applied rewrites 80.4%

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

Alternative 11: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, t\_0, 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_0 \cdot 0.25, {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}, 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)))
   (if (<= d -2.3e+55)
     (*
      (fma (* 0.25 (pow (/ d (* M D)) -2.0)) t_0 1.0)
      (/ (sqrt (- d)) (sqrt (* (- h) (/ l d)))))
     (if (<= d -6.4e-90)
       (*
        (*
         (sqrt (/ d l))
         (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0))
        (sqrt (/ d h)))
       (if (<= d -5e-311)
         (/
          (/
           (fma
            (* (* (* M M) -0.125) (* D D))
            (/ (/ h l) (sqrt (/ l h)))
            (* (sqrt (/ h l)) (* d d)))
           h)
          d)
         (/
          (/ (* (fma (* t_0 0.25) (pow (/ (/ d M) D) -2.0) 1.0) d) (sqrt l))
          (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (h / l) * -0.5;
	double tmp;
	if (d <= -2.3e+55) {
		tmp = fma((0.25 * pow((d / (M * D)), -2.0)), t_0, 1.0) * (sqrt(-d) / sqrt((-h * (l / d))));
	} else if (d <= -6.4e-90) {
		tmp = (sqrt((d / l)) * fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0)) * sqrt((d / h));
	} else if (d <= -5e-311) {
		tmp = (fma((((M * M) * -0.125) * (D * D)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * (d * d))) / h) / d;
	} else {
		tmp = ((fma((t_0 * 0.25), pow(((d / M) / D), -2.0), 1.0) * d) / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(h / l) * -0.5)
	tmp = 0.0
	if (d <= -2.3e+55)
		tmp = Float64(fma(Float64(0.25 * (Float64(d / Float64(M * D)) ^ -2.0)), t_0, 1.0) * Float64(sqrt(Float64(-d)) / sqrt(Float64(Float64(-h) * Float64(l / d)))));
	elseif (d <= -6.4e-90)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0)) * sqrt(Float64(d / h)));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M * M) * -0.125) * Float64(D * D)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / h) / d);
	else
		tmp = Float64(Float64(Float64(fma(Float64(t_0 * 0.25), (Float64(Float64(d / M) / D) ^ -2.0), 1.0) * d) / sqrt(l)) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[d, -2.3e+55], N[(N[(N[(0.25 * N[Power[N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[N[((-h) * N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.4e-90], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 * 0.25), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.29999999999999987e55

   1. Initial program 76.3%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. sqrt-unprod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. frac-times N/A

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

      14. sqrt-div N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-neg.f64 N/A

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

      22. lower-/.f64 86.7

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

   8. Applied rewrites 86.7%

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

   if -2.29999999999999987e55 < d < -6.40000000000000014e-90

   1. Initial program 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 93.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 93.4

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 93.4

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

   6. Applied rewrites 93.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 93.4

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 93.4

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

   8. Applied rewrites 93.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 93.4%

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

   if -6.40000000000000014e-90 < d < -5.00000000000023e-311

   1. Initial program 45.0%

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

   3. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot M\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 47.7%

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

   10. Applied rewrites 56.2%

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

   if -5.00000000000023e-311 < d

   1. Initial program 72.1%

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 69.0%

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

   8. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot 0.25, {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}, 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot 0.25, {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}, 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (sqrt (/ d l))
           (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0))
          (sqrt (/ d h)))))
   (if (<= d -6.4e-90)
     t_0
     (if (<= d -5e-311)
       (/
        (/
         (fma
          (* (* (* M M) -0.125) (* D D))
          (/ (/ h l) (sqrt (/ l h)))
          (* (sqrt (/ h l)) (* d d)))
         h)
        d)
       (if (<= d 1.45e-42)
         (/
          (fma
           (* (* D D) (* (/ (/ (* M M) d) l) -0.125))
           (sqrt h)
           (* (sqrt (pow h -1.0)) d))
          (sqrt l))
         t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0)) * sqrt((d / h));
	double tmp;
	if (d <= -6.4e-90) {
		tmp = t_0;
	} else if (d <= -5e-311) {
		tmp = (fma((((M * M) * -0.125) * (D * D)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * (d * d))) / h) / d;
	} else if (d <= 1.45e-42) {
		tmp = fma(((D * D) * ((((M * M) / d) / l) * -0.125)), sqrt(h), (sqrt(pow(h, -1.0)) * d)) / sqrt(l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (d <= -6.4e-90)
		tmp = t_0;
	elseif (d <= -5e-311)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M * M) * -0.125) * Float64(D * D)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / h) / d);
	elseif (d <= 1.45e-42)
		tmp = Float64(fma(Float64(Float64(D * D) * Float64(Float64(Float64(Float64(M * M) / d) / l) * -0.125)), sqrt(h), Float64(sqrt((h ^ -1.0)) * d)) / sqrt(l));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.4e-90], t$95$0, If[LessEqual[d, -5e-311], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.45e-42], N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.40000000000000014e-90 or 1.4500000000000001e-42 < d

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   6. Applied rewrites 84.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 84.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 84.6

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

   8. Applied rewrites 84.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 84.6%

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

   if -6.40000000000000014e-90 < d < -5.00000000000023e-311

   1. Initial program 45.0%

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

   3. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot M\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 47.7%

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

   10. Applied rewrites 56.2%

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

   if -5.00000000000023e-311 < d < 1.4500000000000001e-42

   1. Initial program 59.7%

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 70.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)}}{\sqrt{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{h}{\ell} \cdot t\_1, t\_1 \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (sqrt (/ d l))
           (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0))
          (sqrt (/ d h))))
        (t_1 (sqrt (/ h l))))
   (if (<= d -6.4e-90)
     t_0
     (if (<= d -5e-311)
       (/
        (/
         (fma (* (* (* M M) -0.125) (* D D)) (* (/ h l) t_1) (* t_1 (* d d)))
         h)
        d)
       (if (<= d 1.45e-42)
         (/
          (fma
           (* (* D D) (* (/ (/ (* M M) d) l) -0.125))
           (sqrt h)
           (* (sqrt (pow h -1.0)) d))
          (sqrt l))
         t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (sqrt((d / l)) * fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0)) * sqrt((d / h));
	double t_1 = sqrt((h / l));
	double tmp;
	if (d <= -6.4e-90) {
		tmp = t_0;
	} else if (d <= -5e-311) {
		tmp = (fma((((M * M) * -0.125) * (D * D)), ((h / l) * t_1), (t_1 * (d * d))) / h) / d;
	} else if (d <= 1.45e-42) {
		tmp = fma(((D * D) * ((((M * M) / d) / l) * -0.125)), sqrt(h), (sqrt(pow(h, -1.0)) * d)) / sqrt(l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0)) * sqrt(Float64(d / h)))
	t_1 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -6.4e-90)
		tmp = t_0;
	elseif (d <= -5e-311)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M * M) * -0.125) * Float64(D * D)), Float64(Float64(h / l) * t_1), Float64(t_1 * Float64(d * d))) / h) / d);
	elseif (d <= 1.45e-42)
		tmp = Float64(fma(Float64(Float64(D * D) * Float64(Float64(Float64(Float64(M * M) / d) / l) * -0.125)), sqrt(h), Float64(sqrt((h ^ -1.0)) * d)) / sqrt(l));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6.4e-90], t$95$0, If[LessEqual[d, -5e-311], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.45e-42], N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.40000000000000014e-90 or 1.4500000000000001e-42 < d

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 84.7

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

   6. Applied rewrites 84.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 84.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 84.6

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

   8. Applied rewrites 84.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 84.6%

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

   if -6.40000000000000014e-90 < d < -5.00000000000023e-311

   1. Initial program 45.0%

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

   3. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot M\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 47.7%

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

   10. Applied rewrites 56.2%

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

   if -5.00000000000023e-311 < d < 1.4500000000000001e-42

   1. Initial program 59.7%

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 70.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{\frac{1}{h}} \cdot d\right)}}{\sqrt{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{h}{\ell} \cdot t\_0, t\_0 \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= d -1.6e+77)
     (/ (sqrt (- d)) (sqrt (* (- h) (/ l d))))
     (if (<= d -5e-311)
       (/
        (/
         (fma (* (* (* M M) -0.125) (* D D)) (* (/ h l) t_0) (* t_0 (* d d)))
         h)
        d)
       (if (<= d 3.5e-44)
         (/
          (fma
           (* (* D D) (* (/ (/ (* M M) d) l) -0.125))
           (sqrt h)
           (* (sqrt (pow h -1.0)) d))
          (sqrt l))
         (*
          (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0)
          (/ d (sqrt (* l h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (d <= -1.6e+77) {
		tmp = sqrt(-d) / sqrt((-h * (l / d)));
	} else if (d <= -5e-311) {
		tmp = (fma((((M * M) * -0.125) * (D * D)), ((h / l) * t_0), (t_0 * (d * d))) / h) / d;
	} else if (d <= 3.5e-44) {
		tmp = fma(((D * D) * ((((M * M) / d) / l) * -0.125)), sqrt(h), (sqrt(pow(h, -1.0)) * d)) / sqrt(l);
	} else {
		tmp = fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0) * (d / sqrt((l * h)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (d <= -1.6e+77)
		tmp = Float64(sqrt(Float64(-d)) / sqrt(Float64(Float64(-h) * Float64(l / d))));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M * M) * -0.125) * Float64(D * D)), Float64(Float64(h / l) * t_0), Float64(t_0 * Float64(d * d))) / h) / d);
	elseif (d <= 3.5e-44)
		tmp = Float64(fma(Float64(Float64(D * D) * Float64(Float64(Float64(Float64(M * M) / d) / l) * -0.125)), sqrt(h), Float64(sqrt((h ^ -1.0)) * d)) / sqrt(l));
	else
		tmp = Float64(fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0) * Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.6e+77], N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[N[((-h) * N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 3.5e-44], N[(N[(N[(N[(D * D), $MachinePrecision] * N[(N[(N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision] / l), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision] * N[Sqrt[h], $MachinePrecision] + N[(N[Sqrt[N[Power[h, -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.6000000000000001e77

   1. Initial program 76.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 4.4

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 69.4%

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

   if -1.6000000000000001e77 < d < -5.00000000000023e-311

   1. Initial program 61.9%

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

   3. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot M\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 47.2%

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

   10. Applied rewrites 56.7%

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

   if -5.00000000000023e-311 < d < 3.4999999999999998e-44

   1. Initial program 60.7%

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 69.6%

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

   if 3.4999999999999998e-44 < d

   1. Initial program 80.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 85.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 85.7

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

   6. Applied rewrites 85.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 85.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 85.6

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

   8. Applied rewrites 85.6%

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

   10. Applied rewrites 83.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot \left(\frac{\frac{M \cdot M}{d}}{\ell} \cdot -0.125\right), \sqrt{h}, \sqrt{{h}^{-1}} \cdot d\right)}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot 0.25, {\left(\frac{\frac{d}{M}}{D}\right)}^{-2}, 1\right) \cdot d}{\sqrt{\ell}}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.4e-90)
   (*
    (*
     (sqrt (/ d l))
     (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0))
    (sqrt (/ d h)))
   (if (<= d -5e-311)
     (/
      (/
       (fma
        (* (* (* M M) -0.125) (* D D))
        (/ (/ h l) (sqrt (/ l h)))
        (* (sqrt (/ h l)) (* d d)))
       h)
      d)
     (/
      (/
       (* (fma (* (* (/ h l) -0.5) 0.25) (pow (/ (/ d M) D) -2.0) 1.0) d)
       (sqrt l))
      (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.4e-90) {
		tmp = (sqrt((d / l)) * fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0)) * sqrt((d / h));
	} else if (d <= -5e-311) {
		tmp = (fma((((M * M) * -0.125) * (D * D)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * (d * d))) / h) / d;
	} else {
		tmp = ((fma((((h / l) * -0.5) * 0.25), pow(((d / M) / D), -2.0), 1.0) * d) / sqrt(l)) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.4e-90)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0)) * sqrt(Float64(d / h)));
	elseif (d <= -5e-311)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M * M) * -0.125) * Float64(D * D)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / h) / d);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.5) * 0.25), (Float64(Float64(d / M) / D) ^ -2.0), 1.0) * d) / sqrt(l)) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.4e-90], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5e-311], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * 0.25), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] / D), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.40000000000000014e-90

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 82.7

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

   6. Applied rewrites 82.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 82.7

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

   8. Applied rewrites 82.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 82.7%

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

   if -6.40000000000000014e-90 < d < -5.00000000000023e-311

   1. Initial program 45.0%

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

   3. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 2.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot M\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 47.7%

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

   10. Applied rewrites 56.2%

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

   if -5.00000000000023e-311 < d

   1. Initial program 72.1%

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

   4. Applied rewrites 73.7%

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

   6. Applied rewrites 69.0%

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

   8. Applied rewrites 81.0%

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

Alternative 16: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.4 \cdot 10^{-90}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(\left(M \cdot M\right) \cdot -0.125\right) \cdot \left(D \cdot D\right), \frac{\frac{h}{\ell}}{\sqrt{\frac{\ell}{h}}}, \sqrt{\frac{h}{\ell}} \cdot \left(d \cdot d\right)\right)}{h}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.25 \cdot {\left(\frac{d}{M \cdot D}\right)}^{-2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.4e-90)
   (*
    (*
     (sqrt (/ d l))
     (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0))
    (sqrt (/ d h)))
   (if (<= d 4.1e-279)
     (/
      (/
       (fma
        (* (* (* M M) -0.125) (* D D))
        (/ (/ h l) (sqrt (/ l h)))
        (* (sqrt (/ h l)) (* d d)))
       h)
      d)
     (/
      (*
       (fma (* 0.25 (pow (/ d (* M D)) -2.0)) (* (/ h l) -0.5) 1.0)
       (/ d (sqrt h)))
      (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.4e-90) {
		tmp = (sqrt((d / l)) * fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0)) * sqrt((d / h));
	} else if (d <= 4.1e-279) {
		tmp = (fma((((M * M) * -0.125) * (D * D)), ((h / l) / sqrt((l / h))), (sqrt((h / l)) * (d * d))) / h) / d;
	} else {
		tmp = (fma((0.25 * pow((d / (M * D)), -2.0)), ((h / l) * -0.5), 1.0) * (d / sqrt(h))) / sqrt(l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.4e-90)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0)) * sqrt(Float64(d / h)));
	elseif (d <= 4.1e-279)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(M * M) * -0.125) * Float64(D * D)), Float64(Float64(h / l) / sqrt(Float64(l / h))), Float64(sqrt(Float64(h / l)) * Float64(d * d))) / h) / d);
	else
		tmp = Float64(Float64(fma(Float64(0.25 * (Float64(d / Float64(M * D)) ^ -2.0)), Float64(Float64(h / l) * -0.5), 1.0) * Float64(d / sqrt(h))) / sqrt(l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.4e-90], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e-279], N[(N[(N[(N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] / N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(0.25 * N[Power[N[(d / N[(M * D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.40000000000000014e-90

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 82.7

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

   6. Applied rewrites 82.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.7

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 82.7

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

   8. Applied rewrites 82.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 82.7%

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

   if -6.40000000000000014e-90 < d < 4.10000000000000017e-279

   1. Initial program 44.3%

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

   3. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 5.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(D \cdot D\right) \cdot -0.125, \sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(M \cdot M\right), \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(d \cdot d\right)\right)}{d}} \]

   6. Taylor expanded in h around 0

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

   8. Applied rewrites 49.0%

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

   10. Applied rewrites 57.1%

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

   if 4.10000000000000017e-279 < d

   1. Initial program 72.8%

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

   4. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 82.1%

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

Alternative 17: 46.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.45e-277)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (* (pow (* (sqrt l) (sqrt h)) -1.0) d)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.45e-277) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = pow((sqrt(l) * sqrt(h)), -1.0) * d;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 1.45d-277) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = ((sqrt(l) * sqrt(h)) ** (-1.0d0)) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.45e-277) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = Math.pow((Math.sqrt(l) * Math.sqrt(h)), -1.0) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.45e-277:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = math.pow((math.sqrt(l) * math.sqrt(h)), -1.0) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.45e-277)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64((Float64(sqrt(l) * sqrt(h)) ^ -1.0) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.45e-277)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = ((sqrt(l) * sqrt(h)) ^ -1.0) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.45e-277], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.44999999999999989e-277

   1. Initial program 68.6%

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

   3. 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 32.8

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

   5. Applied rewrites 32.8%

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

   if 1.44999999999999989e-277 < l

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 46.5%

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

   9. Applied rewrites 53.3%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\ell} \cdot \sqrt{h}\right)}^{-1} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{-1}{-\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.1e-232)
   (/ (sqrt (- d)) (sqrt (* (- h) (/ l d))))
   (if (<= l 1.45e-277)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (* (/ (sqrt (/ -1.0 (- l))) (sqrt h)) d))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.1e-232) {
		tmp = sqrt(-d) / sqrt((-h * (l / d)));
	} else if (l <= 1.45e-277) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = (sqrt((-1.0 / -l)) / sqrt(h)) * d;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-2.1d-232)) then
        tmp = sqrt(-d) / sqrt((-h * (l / d)))
    else if (l <= 1.45d-277) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = (sqrt(((-1.0d0) / -l)) / sqrt(h)) * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.1e-232) {
		tmp = Math.sqrt(-d) / Math.sqrt((-h * (l / d)));
	} else if (l <= 1.45e-277) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = (Math.sqrt((-1.0 / -l)) / Math.sqrt(h)) * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.1e-232:
		tmp = math.sqrt(-d) / math.sqrt((-h * (l / d)))
	elif l <= 1.45e-277:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = (math.sqrt((-1.0 / -l)) / math.sqrt(h)) * d
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.1e-232)
		tmp = Float64(sqrt(Float64(-d)) / sqrt(Float64(Float64(-h) * Float64(l / d))));
	elseif (l <= 1.45e-277)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(-1.0 / Float64(-l))) / sqrt(h)) * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -2.1e-232)
		tmp = sqrt(-d) / sqrt((-h * (l / d)));
	elseif (l <= 1.45e-277)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = (sqrt((-1.0 / -l)) / sqrt(h)) * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.1e-232], N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[N[((-h) * N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e-277], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(-1.0 / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.1e-232

   1. Initial program 67.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 8.7

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

   5. Applied rewrites 8.7%

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

   7. Applied rewrites 35.8%

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

   if -2.1e-232 < l < 1.44999999999999989e-277

   1. Initial program 75.0%

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

   3. Taylor expanded in 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 50.4

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

   5. Applied rewrites 50.4%

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

   if 1.44999999999999989e-277 < l

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 46.5%

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

   9. Applied rewrites 53.3%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{-1}{-\ell}}}{\sqrt{h}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l 1.45e-277)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.45e-277) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 1.45d-277) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 1.45e-277) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= 1.45e-277:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 1.45e-277)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 1.45e-277)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, 1.45e-277], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.44999999999999989e-277

   1. Initial program 68.6%

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

   3. 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 32.8

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

   5. Applied rewrites 32.8%

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

   if 1.44999999999999989e-277 < l

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5

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

   5. Applied rewrites 46.5%

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

   7. Applied rewrites 46.4%

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

   9. Applied rewrites 53.3%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.45 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot d\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (pow (* l h) -1.0))))
   (if (<= l 1.5e-277) (* (- d) t_0) (* t_0 d))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(pow((l * h), -1.0));
	double tmp;
	if (l <= 1.5e-277) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((l * h) ** (-1.0d0)))
    if (l <= 1.5d-277) then
        tmp = -d * t_0
    else
        tmp = t_0 * d
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(Math.pow((l * h), -1.0));
	double tmp;
	if (l <= 1.5e-277) {
		tmp = -d * t_0;
	} else {
		tmp = t_0 * d;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(math.pow((l * h), -1.0))
	tmp = 0
	if l <= 1.5e-277:
		tmp = -d * t_0
	else:
		tmp = t_0 * d
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt((Float64(l * h) ^ -1.0))
	tmp = 0.0
	if (l <= 1.5e-277)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(t_0 * d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((l * h) ^ -1.0));
	tmp = 0.0;
	if (l <= 1.5e-277)
		tmp = -d * t_0;
	else
		tmp = t_0 * d;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, 1.5e-277], N[((-d) * t$95$0), $MachinePrecision], N[(t$95$0 * d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.49999999999999989e-277

   1. Initial program 68.6%

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

   3. 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 32.8

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

   5. Applied rewrites 32.8%

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

   if 1.49999999999999989e-277 < l

   1. Initial program 70.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.5

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

   5. Applied rewrites 46.5%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-277}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.1% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

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 = sqrt(((l * h) ** (-1.0d0))) * d
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 27.4

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

5. Applied rewrites 27.4%

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

6. Final simplification 27.4%

   \[\leadsto \sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d \]
7. Add Preprocessing

Alternative 22: 55.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot D}{\ell \cdot d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6e-305)
   (/ (sqrt (- d)) (sqrt (* (- h) (/ l d))))
   (if (<= l 7.6e+117)
     (*
      (fma (/ (* -0.5 (* M (/ D d))) l) (/ (* (* (* 0.25 D) M) h) d) 1.0)
      (/ d (sqrt (* l h))))
     (if (<= l 1.8e+237)
       (/ (/ (* (* (* (* (sqrt h) -0.125) M) (* M D)) D) (* l d)) (sqrt l))
       (/ (/ d (sqrt h)) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e-305) {
		tmp = sqrt(-d) / sqrt((-h * (l / d)));
	} else if (l <= 7.6e+117) {
		tmp = fma(((-0.5 * (M * (D / d))) / l), ((((0.25 * D) * M) * h) / d), 1.0) * (d / sqrt((l * h)));
	} else if (l <= 1.8e+237) {
		tmp = (((((sqrt(h) * -0.125) * M) * (M * D)) * D) / (l * d)) / sqrt(l);
	} else {
		tmp = (d / sqrt(h)) / sqrt(l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6e-305)
		tmp = Float64(sqrt(Float64(-d)) / sqrt(Float64(Float64(-h) * Float64(l / d))));
	elseif (l <= 7.6e+117)
		tmp = Float64(fma(Float64(Float64(-0.5 * Float64(M * Float64(D / d))) / l), Float64(Float64(Float64(Float64(0.25 * D) * M) * h) / d), 1.0) * Float64(d / sqrt(Float64(l * h))));
	elseif (l <= 1.8e+237)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(sqrt(h) * -0.125) * M) * Float64(M * D)) * D) / Float64(l * d)) / sqrt(l));
	else
		tmp = Float64(Float64(d / sqrt(h)) / sqrt(l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6e-305], N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[N[((-h) * N[(l / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.6e+117], N[(N[(N[(N[(-0.5 * N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(N[(0.25 * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.8e+237], N[(N[(N[(N[(N[(N[(N[Sqrt[h], $MachinePrecision] * -0.125), $MachinePrecision] * M), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] * D), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.0000000000000002e-305

   1. Initial program 67.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 9.6

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 35.7%

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

   if -6.0000000000000002e-305 < l < 7.6000000000000003e117

   1. Initial program 80.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.0

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*r* N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-*.f64 N/A

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

      21. metadata-eval 82.0

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

   6. Applied rewrites 82.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \color{blue}{\frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. metadata-eval 82.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 82.0

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

   8. Applied rewrites 82.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \frac{\left(M \cdot \left(0.25 \cdot D\right)\right) \cdot h}{d}\right) \]

   10. Applied rewrites 86.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}} \]

   if 7.6000000000000003e117 < l < 1.80000000000000007e237

   1. Initial program 45.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{d}}{\sqrt{\ell}}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \ell} \cdot \sqrt{h}\right)}}{\sqrt{\ell}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \color{blue}{\left(\sqrt{h} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \ell}\right)}}{\sqrt{\ell}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \ell}}}{\sqrt{\ell}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \ell}}}{\sqrt{\ell}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot \sqrt{h}\right)} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \ell}}{\sqrt{\ell}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \color{blue}{\sqrt{h}}\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \ell}}{\sqrt{\ell}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \frac{\color{blue}{{M}^{2} \cdot {D}^{2}}}{d \cdot \ell}}{\sqrt{\ell}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \frac{{M}^{2} \cdot \color{blue}{\left(D \cdot D\right)}}{d \cdot \ell}}{\sqrt{\ell}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \frac{\color{blue}{\left({M}^{2} \cdot D\right) \cdot D}}{d \cdot \ell}}{\sqrt{\ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \frac{\left({M}^{2} \cdot D\right) \cdot D}{\color{blue}{\ell \cdot d}}}{\sqrt{\ell}} \]

      10. times-frac N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \color{blue}{\left(\frac{{M}^{2} \cdot D}{\ell} \cdot \frac{D}{d}\right)}}{\sqrt{\ell}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \color{blue}{\left(\frac{{M}^{2} \cdot D}{\ell} \cdot \frac{D}{d}\right)}}{\sqrt{\ell}} \]

      12. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \left(\color{blue}{\frac{{M}^{2} \cdot D}{\ell}} \cdot \frac{D}{d}\right)}{\sqrt{\ell}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \left(\frac{\color{blue}{{M}^{2} \cdot D}}{\ell} \cdot \frac{D}{d}\right)}{\sqrt{\ell}} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot D}{\ell} \cdot \frac{D}{d}\right)}{\sqrt{\ell}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{-1}{8} \cdot \sqrt{h}\right) \cdot \left(\frac{\color{blue}{\left(M \cdot M\right)} \cdot D}{\ell} \cdot \frac{D}{d}\right)}{\sqrt{\ell}} \]

      16. lower-/.f64 37.1

          \[\leadsto \frac{\left(-0.125 \cdot \sqrt{h}\right) \cdot \left(\frac{\left(M \cdot M\right) \cdot D}{\ell} \cdot \color{blue}{\frac{D}{d}}\right)}{\sqrt{\ell}} \]

   7. Applied rewrites 37.1%

      \[\leadsto \frac{\color{blue}{\left(-0.125 \cdot \sqrt{h}\right) \cdot \left(\frac{\left(M \cdot M\right) \cdot D}{\ell} \cdot \frac{D}{d}\right)}}{\sqrt{\ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.9%

      \[\leadsto \frac{\left(-0.125 \cdot \sqrt{h}\right) \cdot \left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \color{blue}{\frac{D}{\ell \cdot d}}\right)}{\sqrt{\ell}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.1%

       \[\leadsto \frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot D}{\color{blue}{\ell \cdot d}}}{\sqrt{\ell}} \]

   if 1.80000000000000007e237 < l

   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 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 64.2

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   5. Applied rewrites 64.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.4%

      \[\leadsto \frac{\frac{d}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{\left(-h\right) \cdot \frac{\ell}{d}}}\\ \mathbf{elif}\;\ell \leq 7.6 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5 \cdot \left(M \cdot \frac{D}{d}\right)}{\ell}, \frac{\left(\left(0.25 \cdot D\right) \cdot M\right) \cdot h}{d}, 1\right) \cdot \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(\sqrt{h} \cdot -0.125\right) \cdot M\right) \cdot \left(M \cdot D\right)\right) \cdot D}{\ell \cdot d}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 25.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

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 / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 27.4

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

5. Applied rewrites 27.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
6. Step-by-step derivation

7. Applied rewrites 27.1%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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)))))