Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.9% → 78.2%

Time: 20.6s

Alternatives: 25

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 65.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot D\right)\\ t_1 := 1 - \frac{t\_0}{d \cdot 2} \cdot \frac{\frac{t\_0}{d} \cdot h}{\ell}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-105}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\frac{\sqrt{\ell}}{\sqrt{d}}}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, \ell \cdot \sqrt{\frac{\ell}{h}}, \sqrt{h \cdot \ell} \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* M D)))
        (t_1 (- 1.0 (* (/ t_0 (* d 2.0)) (/ (* (/ t_0 d) h) l)))))
   (if (<= l -5e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (/ 1.0 2.0))) t_1)
     (if (<= l 2.7e-105)
       (* (* (pow (/ d h) (/ 1.0 2.0)) (/ 1.0 (/ (sqrt l) (sqrt d)))) t_1)
       (if (<= l 1.05e+104)
         (/
          (/
           (fma
            d
            (* l (sqrt (/ l h)))
            (* (sqrt (* h l)) (* -0.125 (/ (* (* M D) (* M D)) d))))
           l)
          l)
         (* d (/ (/ 1.0 (sqrt h)) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (M * D);
	double t_1 = 1.0 - ((t_0 / (d * 2.0)) * (((t_0 / d) * h) / l));
	double tmp;
	if (l <= -5e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), (1.0 / 2.0))) * t_1;
	} else if (l <= 2.7e-105) {
		tmp = (pow((d / h), (1.0 / 2.0)) * (1.0 / (sqrt(l) / sqrt(d)))) * t_1;
	} else if (l <= 1.05e+104) {
		tmp = (fma(d, (l * sqrt((l / h))), (sqrt((h * l)) * (-0.125 * (((M * D) * (M * D)) / d)))) / l) / l;
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(M * D))
	t_1 = Float64(1.0 - Float64(Float64(t_0 / Float64(d * 2.0)) * Float64(Float64(Float64(t_0 / d) * h) / l)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_1);
	elseif (l <= 2.7e-105)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(1.0 / Float64(sqrt(l) / sqrt(d)))) * t_1);
	elseif (l <= 1.05e+104)
		tmp = Float64(Float64(fma(d, Float64(l * sqrt(Float64(l / h))), Float64(sqrt(Float64(h * l)) * Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / d)))) / l) / l);
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / d), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 2.7e-105], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.05e+104], N[(N[(N[(d * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 63.6%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 68.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 81.1

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

   6. Applied rewrites 81.1%

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

   if -4.999999999999985e-310 < l < 2.69999999999999993e-105

   1. Initial program 73.2%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 78.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 87.7

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

   6. Applied rewrites 87.7%

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

   if 2.69999999999999993e-105 < l < 1.0499999999999999e104

   1. Initial program 71.2%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 63.1%

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

   9. Applied rewrites 94.3%

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

   if 1.0499999999999999e104 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 68.6

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 79.9%

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

Alternative 2: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{t\_0 \cdot \left(\frac{h \cdot \left(D \cdot D\right)}{d \cdot \left(d \cdot \ell\right)} \cdot \left(-0.125 \cdot \left(M \cdot M\right)\right)\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(M, \left(M \cdot D\right) \cdot \frac{\left(D \cdot -0.5\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot -2\right) \cdot \ell}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (fabs (/ d (sqrt (* h l)))))
        (t_2
         (*
          (* (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))))))
   (if (<= t_2 (- INFINITY))
     (/
      (* t_0 (* (/ (* h (* D D)) (* d (* d l))) (* -0.125 (* M M))))
      (sqrt (/ h d)))
     (if (<= t_2 -2e-94)
       (*
        (fma
         M
         (* (* M D) (/ (* (* D -0.5) (* h -0.5)) (* (* (* d d) -2.0) l)))
         1.0)
        (sqrt (/ (* d d) (* h l))))
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 5e+286) (* t_0 (sqrt (/ d h))) t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = fabs((d / sqrt((h * l))));
	double t_2 = (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)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (t_0 * (((h * (D * D)) / (d * (d * l))) * (-0.125 * (M * M)))) / sqrt((h / d));
	} else if (t_2 <= -2e-94) {
		tmp = fma(M, ((M * D) * (((D * -0.5) * (h * -0.5)) / (((d * d) * -2.0) * l))), 1.0) * sqrt(((d * d) / (h * l)));
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+286) {
		tmp = t_0 * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = abs(Float64(d / sqrt(Float64(h * l))))
	t_2 = 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))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * Float64(Float64(Float64(h * Float64(D * D)) / Float64(d * Float64(d * l))) * Float64(-0.125 * Float64(M * M)))) / sqrt(Float64(h / d)));
	elseif (t_2 <= -2e-94)
		tmp = Float64(fma(M, Float64(Float64(M * D) * Float64(Float64(Float64(D * -0.5) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * -2.0) * l))), 1.0) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+286)
		tmp = Float64(t_0 * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(t$95$0 * N[(N[(N[(h * N[(D * D), $MachinePrecision]), $MachinePrecision] / N[(d * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-94], N[(N[(M * N[(N[(M * D), $MachinePrecision] * N[(N[(N[(D * -0.5), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * -2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e+286], N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 81.4%

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in M around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-94

   1. Initial program 98.2%

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

   4. Applied rewrites 53.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. unpow-prod-down N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

   6. Applied rewrites 61.7%

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

   8. Applied rewrites 58.1%

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

   if -1.9999999999999999e-94 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 26.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 31.6

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

   5. Applied rewrites 31.6%

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

   7. Applied rewrites 31.6%

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

   9. Applied rewrites 60.3%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 3: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := 0.25 \cdot \frac{M \cdot D}{d}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot t\_0\right) \cdot \left(1 - \frac{t\_1}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot t\_1}{d}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (* 0.25 (/ (* M D) d)))
        (t_2
         (*
          (* (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)))))
        (t_3 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_2 -5e-250)
     (*
      (* (sqrt (/ d h)) t_0)
      (- 1.0 (/ t_1 (* (/ l (* h (* 0.5 (* M D)))) d))))
     (if (<= t_2 0.0)
       t_3
       (if (<= t_2 5e+286)
         (/
          (* t_0 (fma (/ (* (* M D) t_1) d) (* -0.5 (/ h l)) 1.0))
          (sqrt (/ h d)))
         t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = 0.25 * ((M * D) / d);
	double t_2 = (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)));
	double t_3 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_2 <= -5e-250) {
		tmp = (sqrt((d / h)) * t_0) * (1.0 - (t_1 / ((l / (h * (0.5 * (M * D)))) * d)));
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 5e+286) {
		tmp = (t_0 * fma((((M * D) * t_1) / d), (-0.5 * (h / l)), 1.0)) / sqrt((h / d));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(0.25 * Float64(Float64(M * D) / d))
	t_2 = 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))))
	t_3 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_2 <= -5e-250)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * t_0) * Float64(1.0 - Float64(t_1 / Float64(Float64(l / Float64(h * Float64(0.5 * Float64(M * D)))) * d))));
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 5e+286)
		tmp = Float64(Float64(t_0 * fma(Float64(Float64(Float64(M * D) * t_1) / d), Float64(-0.5 * Float64(h / l)), 1.0)) / sqrt(Float64(h / d)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e-250], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 - N[(t$95$1 / N[(N[(l / N[(h * N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 5e+286], N[(N[(t$95$0 * N[(N[(N[(N[(M * D), $MachinePrecision] * t$95$1), $MachinePrecision] / d), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000027e-250

   1. Initial program 83.6%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 87.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 87.3

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

   8. Applied rewrites 87.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.3

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 87.3

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

   10. Applied rewrites 87.3%

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

   if -5.00000000000000027e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 25.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 32.1%

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

   9. Applied rewrites 61.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 98.6%

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

Alternative 4: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\left(t\_0 \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (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)))))
        (t_2 (sqrt (/ d h)))
        (t_3 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_1 -5e-250)
     (*
      (*
       t_0
       (fma (/ (* M (* D (* M D))) (* (* d d) 4.0)) (* -0.5 (/ h l)) 1.0))
      t_2)
     (if (<= t_1 0.0) t_3 (if (<= t_1 5e+286) (* t_0 t_2) t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = (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)));
	double t_2 = sqrt((d / h));
	double t_3 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = (t_0 * fma(((M * (D * (M * D))) / ((d * d) * 4.0)), (-0.5 * (h / l)), 1.0)) * t_2;
	} else if (t_1 <= 0.0) {
		tmp = t_3;
	} else if (t_1 <= 5e+286) {
		tmp = t_0 * t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = 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))))
	t_2 = sqrt(Float64(d / h))
	t_3 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = Float64(Float64(t_0 * fma(Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(Float64(d * d) * 4.0)), Float64(-0.5 * Float64(h / l)), 1.0)) * t_2);
	elseif (t_1 <= 0.0)
		tmp = t_3;
	elseif (t_1 <= 5e+286)
		tmp = Float64(t_0 * t_2);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-250], N[(N[(t$95$0 * N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$3, If[LessEqual[t$95$1, 5e+286], N[(t$95$0 * t$95$2), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000027e-250

   1. Initial program 83.6%

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

   4. Applied rewrites 64.8%

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

   if -5.00000000000000027e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 25.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 32.1%

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

   9. Applied rewrites 61.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 5: 70.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\left(\mathsf{fma}\left(M, \left(M \cdot D\right) \cdot \frac{\left(D \cdot -0.5\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot -2\right) \cdot \ell}, 1\right) \cdot t\_2\right) \cdot t\_3\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_2 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (* (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)))))
        (t_2 (sqrt (/ d l)))
        (t_3 (sqrt (/ d h))))
   (if (<= t_1 -5e-250)
     (*
      (*
       (fma
        M
        (* (* M D) (/ (* (* D -0.5) (* h -0.5)) (* (* (* d d) -2.0) l)))
        1.0)
       t_2)
      t_3)
     (if (<= t_1 0.0) t_0 (if (<= t_1 5e+286) (* t_2 t_3) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (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)));
	double t_2 = sqrt((d / l));
	double t_3 = sqrt((d / h));
	double tmp;
	if (t_1 <= -5e-250) {
		tmp = (fma(M, ((M * D) * (((D * -0.5) * (h * -0.5)) / (((d * d) * -2.0) * l))), 1.0) * t_2) * t_3;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+286) {
		tmp = t_2 * t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = 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))))
	t_2 = sqrt(Float64(d / l))
	t_3 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_1 <= -5e-250)
		tmp = Float64(Float64(fma(M, Float64(Float64(M * D) * Float64(Float64(Float64(D * -0.5) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * -2.0) * l))), 1.0) * t_2) * t_3);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+286)
		tmp = Float64(t_2 * t_3);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-250], N[(N[(N[(M * N[(N[(M * D), $MachinePrecision] * N[(N[(N[(D * -0.5), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * -2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+286], N[(t$95$2 * t$95$3), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000027e-250

   1. Initial program 83.6%

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

   4. Applied rewrites 60.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. unpow-prod-down N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

   6. Applied rewrites 61.3%

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

   8. Applied rewrites 64.5%

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

   if -5.00000000000000027e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 25.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 32.1%

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

   9. Applied rewrites 61.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 6: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\left(t\_2 \cdot \left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right)\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l)))
        (t_3 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_0 -5e-250)
     (*
      (*
       t_2
       (+ 1.0 (/ (* (* D (* M (* M D))) (* h -0.5)) (* (* (* d d) 4.0) l))))
      t_1)
     (if (<= t_0 0.0) t_3 (if (<= t_0 5e+286) (* t_2 t_1) t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double t_3 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-250) {
		tmp = (t_2 * (1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l)))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_3;
	} else if (t_0 <= 5e+286) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (((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)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    t_3 = abs((d / sqrt((h * l))))
    if (t_0 <= (-5d-250)) then
        tmp = (t_2 * (1.0d0 + (((d_1 * (m * (m * d_1))) * (h * (-0.5d0))) / (((d * d) * 4.0d0) * l)))) * t_1
    else if (t_0 <= 0.0d0) then
        tmp = t_3
    else if (t_0 <= 5d+286) then
        tmp = t_2 * t_1
    else
        tmp = t_3
    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.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)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double t_3 = Math.abs((d / Math.sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-250) {
		tmp = (t_2 * (1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l)))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_3;
	} else if (t_0 <= 5e+286) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	t_3 = math.fabs((d / math.sqrt((h * l))))
	tmp = 0
	if t_0 <= -5e-250:
		tmp = (t_2 * (1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l)))) * t_1
	elif t_0 <= 0.0:
		tmp = t_3
	elif t_0 <= 5e+286:
		tmp = t_2 * t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	t_3 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_0 <= -5e-250)
		tmp = Float64(Float64(t_2 * Float64(1.0 + Float64(Float64(Float64(D * Float64(M * Float64(M * D))) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * 4.0) * l)))) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_3;
	elseif (t_0 <= 5e+286)
		tmp = Float64(t_2 * t_1);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	t_3 = abs((d / sqrt((h * l))));
	tmp = 0.0;
	if (t_0 <= -5e-250)
		tmp = (t_2 * (1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l)))) * t_1;
	elseif (t_0 <= 0.0)
		tmp = t_3;
	elseif (t_0 <= 5e+286)
		tmp = t_2 * t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-250], N[(N[(t$95$2 * N[(1.0 + N[(N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$3, If[LessEqual[t$95$0, 5e+286], N[(t$95$2 * t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000027e-250

   1. Initial program 83.6%

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

   4. Applied rewrites 60.1%

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

   6. Applied rewrites 63.4%

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

   if -5.00000000000000027e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 25.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 32.1%

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

   9. Applied rewrites 61.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 7: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_0 -5e+54)
     (*
      (fma (/ (* M (* D (* M D))) (* (* d d) 4.0)) (* -0.5 (/ h l)) 1.0)
      (sqrt (/ (* d d) (* h l))))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 5e+286) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e+54) {
		tmp = fma(((M * (D * (M * D))) / ((d * d) * 4.0)), (-0.5 * (h / l)), 1.0) * sqrt(((d * d) / (h * l)));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_0 <= -5e+54)
		tmp = Float64(fma(Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(Float64(d * d) * 4.0)), Float64(-0.5 * Float64(h / l)), 1.0) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e+54], N[(N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000005e54

   1. Initial program 83.4%

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

   4. Applied rewrites 42.9%

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

   if -5.00000000000000005e54 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 27.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 31.4%

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

   9. Applied rewrites 59.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 8: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(M, \left(M \cdot D\right) \cdot \frac{\left(D \cdot -0.5\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot -2\right) \cdot \ell}, 1\right) \cdot \sqrt{\frac{d \cdot d}{h \cdot \ell}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fabs (/ d (sqrt (* h l)))))
        (t_1
         (*
          (* (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))))))
   (if (<= t_1 -2e-94)
     (*
      (fma
       M
       (* (* M D) (/ (* (* D -0.5) (* h -0.5)) (* (* (* d d) -2.0) l)))
       1.0)
      (sqrt (/ (* d d) (* h l))))
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 5e+286) (* (sqrt (/ d l)) (sqrt (/ d h))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fabs((d / sqrt((h * l))));
	double t_1 = (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)));
	double tmp;
	if (t_1 <= -2e-94) {
		tmp = fma(M, ((M * D) * (((D * -0.5) * (h * -0.5)) / (((d * d) * -2.0) * l))), 1.0) * sqrt(((d * d) / (h * l)));
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = abs(Float64(d / sqrt(Float64(h * l))))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= -2e-94)
		tmp = Float64(fma(M, Float64(Float64(M * D) * Float64(Float64(Float64(D * -0.5) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * -2.0) * l))), 1.0) * sqrt(Float64(Float64(d * d) / Float64(h * l))));
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -2e-94], N[(N[(M * N[(N[(M * D), $MachinePrecision] * N[(N[(N[(D * -0.5), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * -2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d * d), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-94

   1. Initial program 83.6%

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

   4. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. unpow-prod-down N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. pow2 N/A

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

      20. lift-/.f64 N/A

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

      21. associate-*r/ N/A

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

   6. Applied rewrites 62.8%

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

   8. Applied rewrites 42.1%

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

   if -1.9999999999999999e-94 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 26.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 31.6

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

   5. Applied rewrites 31.6%

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

   7. Applied rewrites 31.6%

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

   9. Applied rewrites 60.3%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 9: 59.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \left|t\_0\right|\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{h}{\ell}, \frac{\left(M \cdot D\right) \cdot \left(\left(M \cdot D\right) \cdot 0.125\right)}{d \cdot d}, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l))))
        (t_1
         (*
          (* (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)))))
        (t_2 (fabs t_0)))
   (if (<= t_1 -5e+54)
     (* (fma (- (/ h l)) (/ (* (* M D) (* (* M D) 0.125)) (* d d)) 1.0) t_0)
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 5e+286) (* (sqrt (/ d l)) (sqrt (/ d h))) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double t_1 = (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)));
	double t_2 = fabs(t_0);
	double tmp;
	if (t_1 <= -5e+54) {
		tmp = fma(-(h / l), (((M * D) * ((M * D) * 0.125)) / (d * d)), 1.0) * t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	t_1 = 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))))
	t_2 = abs(t_0)
	tmp = 0.0
	if (t_1 <= -5e+54)
		tmp = Float64(fma(Float64(-Float64(h / l)), Float64(Float64(Float64(M * D) * Float64(Float64(M * D) * 0.125)) / Float64(d * d)), 1.0) * t_0);
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Abs[t$95$0], $MachinePrecision]}, If[LessEqual[t$95$1, -5e+54], N[(N[((-N[(h / l), $MachinePrecision]) * N[(N[(N[(M * D), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000005e54

   1. Initial program 83.4%

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

   4. Applied rewrites 35.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 39.2%

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

   if -5.00000000000000005e54 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 27.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 31.4

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

   5. Applied rewrites 31.4%

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

   7. Applied rewrites 31.4%

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

   9. Applied rewrites 59.8%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 10: 58.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{h \cdot \ell}}\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+196}:\\ \;\;\;\;\left(1 + \frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (/ d (sqrt (* h l))))
        (t_2 (fabs t_1)))
   (if (<= t_0 -1e+196)
     (*
      (+ 1.0 (/ (* (* M (* M (* D D))) (* h -0.5)) (* (* 4.0 (* d d)) l)))
      t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 5e+286) (* (sqrt (/ d l)) (sqrt (/ d h))) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = d / sqrt((h * l));
	double t_2 = fabs(t_1);
	double tmp;
	if (t_0 <= -1e+196) {
		tmp = (1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((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)))
    t_1 = d / sqrt((h * l))
    t_2 = abs(t_1)
    if (t_0 <= (-1d+196)) then
        tmp = (1.0d0 + (((m * (m * (d_1 * d_1))) * (h * (-0.5d0))) / ((4.0d0 * (d * d)) * l))) * t_1
    else if (t_0 <= 0.0d0) then
        tmp = t_2
    else if (t_0 <= 5d+286) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_2
    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.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)));
	double t_1 = d / Math.sqrt((h * l));
	double t_2 = Math.abs(t_1);
	double tmp;
	if (t_0 <= -1e+196) {
		tmp = (1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 5e+286) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	t_1 = d / math.sqrt((h * l))
	t_2 = math.fabs(t_1)
	tmp = 0
	if t_0 <= -1e+196:
		tmp = (1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * t_1
	elif t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 5e+286:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = Float64(d / sqrt(Float64(h * l)))
	t_2 = abs(t_1)
	tmp = 0.0
	if (t_0 <= -1e+196)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(M * Float64(M * Float64(D * D))) * Float64(h * -0.5)) / Float64(Float64(4.0 * Float64(d * d)) * l))) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = d / sqrt((h * l));
	t_2 = abs(t_1);
	tmp = 0.0;
	if (t_0 <= -1e+196)
		tmp = (1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * t_1;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 5e+286)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[t$95$0, -1e+196], N[(N[(1.0 + N[(N[(N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999995e195

   1. Initial program 82.8%

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

   4. Applied rewrites 28.4%

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

   6. Applied rewrites 35.2%

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

   if -9.9999999999999995e195 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 29.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 30.5

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 30.5%

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

   9. Applied rewrites 58.1%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 11: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_0 -5e-250)
     (* (* D (* D (* M M))) (* (sqrt (/ h (* l (* l l)))) (/ -0.125 d)))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 5e+286) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-250) {
		tmp = (D * (D * (M * M))) * (sqrt((h / (l * (l * l)))) * (-0.125 / d));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (((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)))
    t_1 = abs((d / sqrt((h * l))))
    if (t_0 <= (-5d-250)) then
        tmp = (d_1 * (d_1 * (m * m))) * (sqrt((h / (l * (l * l)))) * ((-0.125d0) / d))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+286) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_1
    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.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)));
	double t_1 = Math.abs((d / Math.sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-250) {
		tmp = (D * (D * (M * M))) * (Math.sqrt((h / (l * (l * l)))) * (-0.125 / d));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+286) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	t_1 = math.fabs((d / math.sqrt((h * l))))
	tmp = 0
	if t_0 <= -5e-250:
		tmp = (D * (D * (M * M))) * (math.sqrt((h / (l * (l * l)))) * (-0.125 / d))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+286:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_0 <= -5e-250)
		tmp = Float64(Float64(D * Float64(D * Float64(M * M))) * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(-0.125 / d)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = abs((d / sqrt((h * l))));
	tmp = 0.0;
	if (t_0 <= -5e-250)
		tmp = (D * (D * (M * M))) * (sqrt((h / (l * (l * l)))) * (-0.125 / d));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+286)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-250], N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000027e-250

   1. Initial program 83.6%

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

   3. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. cube-mult N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 N/A

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

   5. Applied rewrites 35.5%

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

   if -5.00000000000000027e-250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 25.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 32.1

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

   5. Applied rewrites 32.1%

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

   7. Applied rewrites 32.1%

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

   9. Applied rewrites 61.5%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 12: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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)))))
        (t_1 (fabs (/ d (sqrt (* h l))))))
   (if (<= t_0 -5e-136)
     (* (sqrt (/ h (* l (* l l)))) (* (* D (* D (* M M))) (/ 0.125 d)))
     (if (<= t_0 0.0)
       t_1
       (if (<= t_0 5e+286) (* (sqrt (/ d l)) (sqrt (/ d h))) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double t_1 = fabs((d / sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-136) {
		tmp = sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (((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)))
    t_1 = abs((d / sqrt((h * l))))
    if (t_0 <= (-5d-136)) then
        tmp = sqrt((h / (l * (l * l)))) * ((d_1 * (d_1 * (m * m))) * (0.125d0 / d))
    else if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+286) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = t_1
    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.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)));
	double t_1 = Math.abs((d / Math.sqrt((h * l))));
	double tmp;
	if (t_0 <= -5e-136) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+286) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	t_1 = math.fabs((d / math.sqrt((h * l))))
	tmp = 0
	if t_0 <= -5e-136:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d))
	elif t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+286:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = t_1
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	t_1 = abs(Float64(d / sqrt(Float64(h * l))))
	tmp = 0.0
	if (t_0 <= -5e-136)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * Float64(D * Float64(M * M))) * Float64(0.125 / d)));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	t_1 = abs((d / sqrt((h * l))));
	tmp = 0.0;
	if (t_0 <= -5e-136)
		tmp = sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	elseif (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+286)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-136], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.0000000000000002e-136

   1. Initial program 83.8%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. associate-*r/ N/A

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

      14. distribute-neg-frac N/A

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

   5. Applied rewrites 27.7%

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

   if -5.0000000000000002e-136 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.0000000000000004e286 < (*.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 26.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 31.8

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

   5. Applied rewrites 31.8%

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

   7. Applied rewrites 31.8%

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

   9. Applied rewrites 60.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

Alternative 13: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(h \cdot \ell\right) \cdot \left(h \cdot \ell\right)}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (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))))))
   (if (<= t_0 0.0)
     (/ d (sqrt (sqrt (* (* h l) (* h l)))))
     (if (<= t_0 5e+286)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (fabs (/ d (sqrt (* h l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (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)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = d / sqrt(sqrt(((h * l) * (h * l))));
	} else if (t_0 <= 5e+286) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	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 = (((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)))
    if (t_0 <= 0.0d0) then
        tmp = d / sqrt(sqrt(((h * l) * (h * l))))
    else if (t_0 <= 5d+286) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs((d / sqrt((h * l))))
    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.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)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = d / Math.sqrt(Math.sqrt(((h * l) * (h * l))));
	} else if (t_0 <= 5e+286) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = d / math.sqrt(math.sqrt(((h * l) * (h * l))))
	elif t_0 <= 5e+286:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = 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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(h * l) * Float64(h * l)))));
	elseif (t_0 <= 5e+286)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (((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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = d / sqrt(sqrt(((h * l) * (h * l))));
	elseif (t_0 <= 5e+286)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(d / N[Sqrt[N[Sqrt[N[(N[(h * l), $MachinePrecision] * N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+286], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 75.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 16.0

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

   5. Applied rewrites 16.0%

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

   7. Applied rewrites 16.0%

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

   9. Applied rewrites 21.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000004e286

   1. Initial program 98.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 97.7%

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

   if 5.0000000000000004e286 < (*.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 23.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 28.2

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

   5. Applied rewrites 28.2%

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

   7. Applied rewrites 28.2%

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

   9. Applied rewrites 58.7%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (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))))
      5e+286)
   (*
    (* (sqrt (/ d h)) (sqrt (/ d l)))
    (- 1.0 (/ (* 0.25 (/ (* M D) d)) (* (/ l (* h (* 0.5 (* M D)))) d))))
   (fabs (/ d (sqrt (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= 5e+286) {
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * (0.5 * (M * D)))) * d)));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	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) ** (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)))) <= 5d+286) then
        tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0d0 - ((0.25d0 * ((m * d_1) / d)) / ((l / (h * (0.5d0 * (m * d_1)))) * d)))
    else
        tmp = abs((d / sqrt((h * l))))
    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), (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)))) <= 5e+286) {
		tmp = (Math.sqrt((d / h)) * Math.sqrt((d / l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * (0.5 * (M * D)))) * d)));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((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)))) <= 5e+286:
		tmp = (math.sqrt((d / h)) * math.sqrt((d / l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * (0.5 * (M * D)))) * d)))
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (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)))) <= 5e+286)
		tmp = Float64(Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(M * D) / d)) / Float64(Float64(l / Float64(h * Float64(0.5 * Float64(M * D)))) * d))));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((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)))) <= 5e+286)
		tmp = (sqrt((d / h)) * sqrt((d / l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * (0.5 * (M * D)))) * d)));
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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], 5e+286], N[(N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(N[(l / N[(h * N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < 5.0000000000000004e286

   1. Initial program 86.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 89.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.8

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 86.8

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

   8. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 86.8

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 86.8

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

   10. Applied rewrites 86.8%

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

   if 5.0000000000000004e286 < (*.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 23.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 28.2

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

   5. Applied rewrites 28.2%

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

   7. Applied rewrites 28.2%

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

   9. Applied rewrites 58.7%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-94}:\\ \;\;\;\;\frac{d}{\sqrt{\sqrt{\left(h \cdot \ell\right) \cdot \left(h \cdot \ell\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{h \cdot \ell}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (* (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))))
      -2e-94)
   (/ d (sqrt (sqrt (* (* h l) (* h l)))))
   (fabs (/ d (sqrt (* h l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((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)))) <= -2e-94) {
		tmp = d / sqrt(sqrt(((h * l) * (h * l))));
	} else {
		tmp = fabs((d / sqrt((h * l))));
	}
	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) ** (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)))) <= (-2d-94)) then
        tmp = d / sqrt(sqrt(((h * l) * (h * l))))
    else
        tmp = abs((d / sqrt((h * l))))
    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), (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)))) <= -2e-94) {
		tmp = d / Math.sqrt(Math.sqrt(((h * l) * (h * l))));
	} else {
		tmp = Math.abs((d / Math.sqrt((h * l))));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((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)))) <= -2e-94:
		tmp = d / math.sqrt(math.sqrt(((h * l) * (h * l))))
	else:
		tmp = math.fabs((d / math.sqrt((h * l))))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (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)))) <= -2e-94)
		tmp = Float64(d / sqrt(sqrt(Float64(Float64(h * l) * Float64(h * l)))));
	else
		tmp = abs(Float64(d / sqrt(Float64(h * l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((((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)))) <= -2e-94)
		tmp = d / sqrt(sqrt(((h * l) * (h * l))));
	else
		tmp = abs((d / sqrt((h * l))));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[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], -2e-94], N[(d / N[Sqrt[N[Sqrt[N[(N[(h * l), $MachinePrecision] * N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $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)))) < -1.9999999999999999e-94

   1. Initial program 83.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 8.6

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

   5. Applied rewrites 8.6%

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

   7. Applied rewrites 8.6%

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

   9. Applied rewrites 16.0%

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

   if -1.9999999999999999e-94 < (*.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 59.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

   7. Applied rewrites 38.6%

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

   9. Applied rewrites 67.6%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{+54}:\\ \;\;\;\;\frac{d}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{t\_0}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<=
        (*
         (* (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))))
        -5e+54)
     (/ d (- t_0))
     (fabs (/ d t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (((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)))) <= -5e+54) {
		tmp = d / -t_0;
	} else {
		tmp = fabs((d / t_0));
	}
	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((h * l))
    if (((((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)))) <= (-5d+54)) then
        tmp = d / -t_0
    else
        tmp = abs((d / t_0))
    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((h * l));
	double tmp;
	if (((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)))) <= -5e+54) {
		tmp = d / -t_0;
	} else {
		tmp = Math.abs((d / t_0));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((h * l))
	tmp = 0
	if ((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)))) <= -5e+54:
		tmp = d / -t_0
	else:
		tmp = math.fabs((d / t_0))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (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)))) <= -5e+54)
		tmp = Float64(d / Float64(-t_0));
	else
		tmp = abs(Float64(d / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((h * l));
	tmp = 0.0;
	if (((((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)))) <= -5e+54)
		tmp = d / -t_0;
	else
		tmp = abs((d / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e+54], N[(d / (-t$95$0)), $MachinePrecision], N[Abs[N[(d / t$95$0), $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)))) < -5.00000000000000005e54

   1. Initial program 83.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 8.6

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

   5. Applied rewrites 8.6%

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

   7. Applied rewrites 8.6%

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

   9. Applied rewrites 8.6%

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

   10. Taylor expanded in h around -inf

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

   12. Applied rewrites 11.1%

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

   if -5.00000000000000005e54 < (*.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 59.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 38.5%

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

   9. Applied rewrites 67.3%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-250}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* h l)))))
   (if (<=
        (*
         (* (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))))
        -5e-250)
     t_0
     (fabs t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((h * l));
	double tmp;
	if (((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)))) <= -5e-250) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	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 = d / sqrt((h * l))
    if (((((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)))) <= (-5d-250)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((h * l));
	double tmp;
	if (((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)))) <= -5e-250) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / math.sqrt((h * l))
	tmp = 0
	if ((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)))) <= -5e-250:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(h * l)))
	tmp = 0.0
	if (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)))) <= -5e-250)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / sqrt((h * l));
	tmp = 0.0;
	if (((((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)))) <= -5e-250)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -5e-250], t$95$0, N[Abs[t$95$0], $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)))) < -5.00000000000000027e-250

   1. Initial program 83.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 8.6

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

   5. Applied rewrites 8.6%

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

   7. Applied rewrites 8.6%

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

   if -5.00000000000000027e-250 < (*.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 58.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   7. Applied rewrites 39.0%

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

   9. Applied rewrites 68.3%

      \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \left(M \cdot \left(D \cdot D\right)\right)\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_3 := \sqrt{-d}\\ t_4 := 1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\\ \mathbf{if}\;h \leq -1.6 \cdot 10^{+269}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{t\_0 \cdot \left(h \cdot -0.5\right)}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_3}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_2 \cdot \frac{t\_3}{\sqrt{-\ell}}\right) \cdot t\_4\\ \mathbf{elif}\;h \leq 7.5 \cdot 10^{+149}:\\ \;\;\;\;\left(t\_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot \frac{\frac{t\_0}{t\_1}}{d \cdot 8}, \frac{-1}{\ell}, \frac{d}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (* M (* D D))))
        (t_1 (sqrt (* h l)))
        (t_2 (pow (/ d h) (/ 1.0 2.0)))
        (t_3 (sqrt (- d)))
        (t_4
         (- 1.0 (/ (* 0.25 (/ (* M D) d)) (* (/ l (* h (* 0.5 (* M D)))) d)))))
   (if (<= h -1.6e+269)
     (/
      (*
       (* (+ 1.0 (/ (* t_0 (* h -0.5)) (* (* 4.0 (* d d)) l))) (sqrt (/ d l)))
       t_3)
      (sqrt (- h)))
     (if (<= h -2e-310)
       (* (* t_2 (/ t_3 (sqrt (- l)))) t_4)
       (if (<= h 7.5e+149)
         (* (* t_2 (/ (sqrt d) (sqrt l))) t_4)
         (fma (* h (/ (/ t_0 t_1) (* d 8.0))) (/ -1.0 l) (/ d t_1)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (M * (D * D));
	double t_1 = sqrt((h * l));
	double t_2 = pow((d / h), (1.0 / 2.0));
	double t_3 = sqrt(-d);
	double t_4 = 1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * (0.5 * (M * D)))) * d));
	double tmp;
	if (h <= -1.6e+269) {
		tmp = (((1.0 + ((t_0 * (h * -0.5)) / ((4.0 * (d * d)) * l))) * sqrt((d / l))) * t_3) / sqrt(-h);
	} else if (h <= -2e-310) {
		tmp = (t_2 * (t_3 / sqrt(-l))) * t_4;
	} else if (h <= 7.5e+149) {
		tmp = (t_2 * (sqrt(d) / sqrt(l))) * t_4;
	} else {
		tmp = fma((h * ((t_0 / t_1) / (d * 8.0))), (-1.0 / l), (d / t_1));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(M * Float64(D * D)))
	t_1 = sqrt(Float64(h * l))
	t_2 = Float64(d / h) ^ Float64(1.0 / 2.0)
	t_3 = sqrt(Float64(-d))
	t_4 = Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(M * D) / d)) / Float64(Float64(l / Float64(h * Float64(0.5 * Float64(M * D)))) * d)))
	tmp = 0.0
	if (h <= -1.6e+269)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(t_0 * Float64(h * -0.5)) / Float64(Float64(4.0 * Float64(d * d)) * l))) * sqrt(Float64(d / l))) * t_3) / sqrt(Float64(-h)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(t_2 * Float64(t_3 / sqrt(Float64(-l)))) * t_4);
	elseif (h <= 7.5e+149)
		tmp = Float64(Float64(t_2 * Float64(sqrt(d) / sqrt(l))) * t_4);
	else
		tmp = fma(Float64(h * Float64(Float64(t_0 / t_1) / Float64(d * 8.0))), Float64(-1.0 / l), Float64(d / t_1));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 - N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(N[(l / N[(h * N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.6e+269], N[(N[(N[(N[(1.0 + N[(N[(t$95$0 * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(t$95$2 * N[(t$95$3 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[h, 7.5e+149], N[(N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(h * N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(d * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / l), $MachinePrecision] + N[(d / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.6e269

   1. Initial program 17.8%

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

   4. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.4

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

   6. Applied rewrites 77.2%

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

   if -1.6e269 < h < -1.999999999999994e-310

   1. Initial program 69.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 71.9

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 71.9

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

   8. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 80.9

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

   10. Applied rewrites 80.9%

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

   if -1.999999999999994e-310 < h < 7.50000000000000031e149

   1. Initial program 76.5%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 78.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 78.5

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lower-/.f64 84.0

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

   8. Applied rewrites 84.0%

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

   if 7.50000000000000031e149 < h

   1. Initial program 42.2%

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

   4. Applied rewrites 14.7%

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

   5. Taylor expanded in M around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 32.5

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

   7. Applied rewrites 32.5%

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

   9. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(h \cdot \frac{\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\sqrt{h \cdot \ell}}}{d \cdot 8}, \frac{-1}{\ell}, \frac{d}{\sqrt{h \cdot \ell}}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 78.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(M \cdot D\right)\\ t_1 := 1 - \frac{t\_0}{d \cdot 2} \cdot \frac{\frac{t\_0}{d} \cdot h}{\ell}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-105}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, \ell \cdot \sqrt{\frac{\ell}{h}}, \sqrt{h \cdot \ell} \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* M D)))
        (t_1 (- 1.0 (* (/ t_0 (* d 2.0)) (/ (* (/ t_0 d) h) l)))))
   (if (<= l -5e-310)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) (pow (/ d l) (/ 1.0 2.0))) t_1)
     (if (<= l 2.7e-105)
       (* (* (pow (/ d h) (/ 1.0 2.0)) (/ (sqrt d) (sqrt l))) t_1)
       (if (<= l 1.05e+104)
         (/
          (/
           (fma
            d
            (* l (sqrt (/ l h)))
            (* (sqrt (* h l)) (* -0.125 (/ (* (* M D) (* M D)) d))))
           l)
          l)
         (* d (/ (/ 1.0 (sqrt h)) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (M * D);
	double t_1 = 1.0 - ((t_0 / (d * 2.0)) * (((t_0 / d) * h) / l));
	double tmp;
	if (l <= -5e-310) {
		tmp = ((sqrt(-d) / sqrt(-h)) * pow((d / l), (1.0 / 2.0))) * t_1;
	} else if (l <= 2.7e-105) {
		tmp = (pow((d / h), (1.0 / 2.0)) * (sqrt(d) / sqrt(l))) * t_1;
	} else if (l <= 1.05e+104) {
		tmp = (fma(d, (l * sqrt((l / h))), (sqrt((h * l)) * (-0.125 * (((M * D) * (M * D)) / d)))) / l) / l;
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64(M * D))
	t_1 = Float64(1.0 - Float64(Float64(t_0 / Float64(d * 2.0)) * Float64(Float64(Float64(t_0 / d) * h) / l)))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_1);
	elseif (l <= 2.7e-105)
		tmp = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(d) / sqrt(l))) * t_1);
	elseif (l <= 1.05e+104)
		tmp = Float64(Float64(fma(d, Float64(l * sqrt(Float64(l / h))), Float64(sqrt(Float64(h * l)) * Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / d)))) / l) / l);
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(t$95$0 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / d), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 2.7e-105], N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, 1.05e+104], N[(N[(N[(d * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 63.6%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 68.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 68.6

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 81.1

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

   6. Applied rewrites 81.1%

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

   if -4.999999999999985e-310 < l < 2.69999999999999993e-105

   1. Initial program 73.2%

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

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

      19. lower-/.f64 N/A

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

   4. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 78.6

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 87.7

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

   6. Applied rewrites 87.7%

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

   if 2.69999999999999993e-105 < l < 1.0499999999999999e104

   1. Initial program 71.2%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 63.1%

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

   9. Applied rewrites 94.3%

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

   if 1.0499999999999999e104 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 68.6

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

   5. Applied rewrites 68.6%

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

   7. Applied rewrites 79.9%

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

Alternative 20: 77.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := 0.5 \cdot \left(M \cdot D\right)\\ \mathbf{if}\;h \leq -1.6 \cdot 10^{+269}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(t\_1 \cdot \frac{t\_0}{\sqrt{-\ell}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot t\_2} \cdot d}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{t\_2}{d \cdot 2} \cdot \frac{\frac{t\_2}{d} \cdot h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (pow (/ d h) (/ 1.0 2.0)))
        (t_2 (* 0.5 (* M D))))
   (if (<= h -1.6e+269)
     (/
      (*
       (*
        (+ 1.0 (/ (* (* M (* M (* D D))) (* h -0.5)) (* (* 4.0 (* d d)) l)))
        (sqrt (/ d l)))
       t_0)
      (sqrt (- h)))
     (if (<= h -2e-310)
       (*
        (* t_1 (/ t_0 (sqrt (- l))))
        (- 1.0 (/ (* 0.25 (/ (* M D) d)) (* (/ l (* h t_2)) d))))
       (*
        (* t_1 (/ (sqrt d) (sqrt l)))
        (- 1.0 (* (/ t_2 (* d 2.0)) (/ (* (/ t_2 d) h) l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double t_1 = pow((d / h), (1.0 / 2.0));
	double t_2 = 0.5 * (M * D);
	double tmp;
	if (h <= -1.6e+269) {
		tmp = (((1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * sqrt((d / l))) * t_0) / sqrt(-h);
	} else if (h <= -2e-310) {
		tmp = (t_1 * (t_0 / sqrt(-l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * t_2)) * d)));
	} else {
		tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0 - ((t_2 / (d * 2.0)) * (((t_2 / d) * h) / l)));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(-d)
    t_1 = (d / h) ** (1.0d0 / 2.0d0)
    t_2 = 0.5d0 * (m * d_1)
    if (h <= (-1.6d+269)) then
        tmp = (((1.0d0 + (((m * (m * (d_1 * d_1))) * (h * (-0.5d0))) / ((4.0d0 * (d * d)) * l))) * sqrt((d / l))) * t_0) / sqrt(-h)
    else if (h <= (-2d-310)) then
        tmp = (t_1 * (t_0 / sqrt(-l))) * (1.0d0 - ((0.25d0 * ((m * d_1) / d)) / ((l / (h * t_2)) * d)))
    else
        tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0d0 - ((t_2 / (d * 2.0d0)) * (((t_2 / d) * h) / l)))
    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(-d);
	double t_1 = Math.pow((d / h), (1.0 / 2.0));
	double t_2 = 0.5 * (M * D);
	double tmp;
	if (h <= -1.6e+269) {
		tmp = (((1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * Math.sqrt((d / l))) * t_0) / Math.sqrt(-h);
	} else if (h <= -2e-310) {
		tmp = (t_1 * (t_0 / Math.sqrt(-l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * t_2)) * d)));
	} else {
		tmp = (t_1 * (Math.sqrt(d) / Math.sqrt(l))) * (1.0 - ((t_2 / (d * 2.0)) * (((t_2 / d) * h) / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	t_1 = math.pow((d / h), (1.0 / 2.0))
	t_2 = 0.5 * (M * D)
	tmp = 0
	if h <= -1.6e+269:
		tmp = (((1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * math.sqrt((d / l))) * t_0) / math.sqrt(-h)
	elif h <= -2e-310:
		tmp = (t_1 * (t_0 / math.sqrt(-l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * t_2)) * d)))
	else:
		tmp = (t_1 * (math.sqrt(d) / math.sqrt(l))) * (1.0 - ((t_2 / (d * 2.0)) * (((t_2 / d) * h) / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(d / h) ^ Float64(1.0 / 2.0)
	t_2 = Float64(0.5 * Float64(M * D))
	tmp = 0.0
	if (h <= -1.6e+269)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(Float64(Float64(M * Float64(M * Float64(D * D))) * Float64(h * -0.5)) / Float64(Float64(4.0 * Float64(d * d)) * l))) * sqrt(Float64(d / l))) * t_0) / sqrt(Float64(-h)));
	elseif (h <= -2e-310)
		tmp = Float64(Float64(t_1 * Float64(t_0 / sqrt(Float64(-l)))) * Float64(1.0 - Float64(Float64(0.25 * Float64(Float64(M * D) / d)) / Float64(Float64(l / Float64(h * t_2)) * d))));
	else
		tmp = Float64(Float64(t_1 * Float64(sqrt(d) / sqrt(l))) * Float64(1.0 - Float64(Float64(t_2 / Float64(d * 2.0)) * Float64(Float64(Float64(t_2 / d) * h) / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	t_1 = (d / h) ^ (1.0 / 2.0);
	t_2 = 0.5 * (M * D);
	tmp = 0.0;
	if (h <= -1.6e+269)
		tmp = (((1.0 + (((M * (M * (D * D))) * (h * -0.5)) / ((4.0 * (d * d)) * l))) * sqrt((d / l))) * t_0) / sqrt(-h);
	elseif (h <= -2e-310)
		tmp = (t_1 * (t_0 / sqrt(-l))) * (1.0 - ((0.25 * ((M * D) / d)) / ((l / (h * t_2)) * d)));
	else
		tmp = (t_1 * (sqrt(d) / sqrt(l))) * (1.0 - ((t_2 / (d * 2.0)) * (((t_2 / d) * h) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(M * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1.6e+269], N[(N[(N[(N[(1.0 + N[(N[(N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-310], N[(N[(t$95$1 * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.25 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / N[(N[(l / N[(h * t$95$2), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(t$95$2 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$2 / d), $MachinePrecision] * h), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.6e269

   1. Initial program 17.8%

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

   4. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.4

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

   6. Applied rewrites 77.2%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{\left(M \cdot \left(M \cdot \left(D \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(4 \cdot \left(d \cdot d\right)\right) \cdot \ell}\right) \cdot \sqrt{\frac{d}{\ell}}\right)} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.6e269 < h < -1.999999999999994e-310

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{h}{\ell}}\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell}}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell}}\right) \]

   4. Applied rewrites 73.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \color{blue}{\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}}\right) \]

      3. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \color{blue}{\frac{1}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}}\right) \]

      4. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2}}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{d \cdot 2}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot d}}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      10. times-frac N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{2}}{2} \cdot \frac{M \cdot D}{d}}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{\frac{1}{2}}{2} \cdot \frac{M \cdot D}{d}}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      12. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{4}} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \color{blue}{\frac{M \cdot D}{d}}}{\frac{\ell}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d}} \cdot h}}\right) \]

      16. associate-*l/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{\color{blue}{\frac{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot h}{d}}}}\right) \]

      17. associate-/r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\color{blue}{\frac{\ell}{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot h} \cdot d}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\color{blue}{\frac{\ell}{\left(\frac{1}{2} \cdot \left(M \cdot D\right)\right) \cdot h} \cdot d}}\right) \]

   6. Applied rewrites 71.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      2. metadata-eval 71.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      5. lift-sqrt.f64 71.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

   8. Applied rewrites 71.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      3. frac-2neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      4. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\color{blue}{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)}}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}\right) \cdot \left(1 - \frac{\frac{1}{4} \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(\frac{1}{2} \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

      9. lower-neg.f64 80.9

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

   10. Applied rewrites 80.9%

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}}\right) \cdot \left(1 - \frac{0.25 \cdot \frac{M \cdot D}{d}}{\frac{\ell}{h \cdot \left(0.5 \cdot \left(M \cdot D\right)\right)} \cdot d}\right) \]

   if -1.999999999999994e-310 < h

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{h}{\ell}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2} \cdot \left(M \cdot D\right)}}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\color{blue}{\frac{1}{2}} \cdot \left(M \cdot D\right)}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{2 \cdot d}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{\color{blue}{d \cdot 2}} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{h}{\ell}}\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell}}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell}}\right) \]

   4. Applied rewrites 72.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}}\right) \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      2. metadata-eval 72.2

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      4. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      6. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      7. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{{d}^{\frac{1}{2}}}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{{d}^{\frac{1}{2}}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      9. pow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{\sqrt{d}}}{\sqrt{\ell}}\right) \cdot \left(1 - \frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{\frac{1}{2} \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

      11. lower-sqrt.f64 79.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\color{blue}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]

   6. Applied rewrites 79.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}}\right) \cdot \left(1 - \frac{0.5 \cdot \left(M \cdot D\right)}{d \cdot 2} \cdot \frac{\frac{0.5 \cdot \left(M \cdot D\right)}{d} \cdot h}{\ell}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 59.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{\left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right) \cdot d}{t\_0}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}, t\_0 \cdot -0.125, d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}{\ell \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -1.35e-158)
     (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
     (if (<= l -8.2e-293)
       (* (sqrt (/ h (* l (* l l)))) (* (* D (* D (* M M))) (/ 0.125 d)))
       (if (<= l 1.2e-173)
         (/
          (*
           (+ 1.0 (/ (* (* D (* M (* M D))) (* h -0.5)) (* (* (* d d) 4.0) l)))
           d)
          t_0)
         (if (<= l 8.5e+103)
           (/
            (fma
             (/ (* (* M D) (* M D)) d)
             (* t_0 -0.125)
             (* d (* l (sqrt (/ l h)))))
            (* l l))
           (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -1.35e-158) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -8.2e-293) {
		tmp = sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	} else if (l <= 1.2e-173) {
		tmp = ((1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l))) * d) / t_0;
	} else if (l <= 8.5e+103) {
		tmp = fma((((M * D) * (M * D)) / d), (t_0 * -0.125), (d * (l * sqrt((l / h))))) / (l * l);
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -1.35e-158)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -8.2e-293)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * Float64(D * Float64(M * M))) * Float64(0.125 / d)));
	elseif (l <= 1.2e-173)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(D * Float64(M * Float64(M * D))) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * 4.0) * l))) * d) / t_0);
	elseif (l <= 8.5e+103)
		tmp = Float64(fma(Float64(Float64(Float64(M * D) * Float64(M * D)) / d), Float64(t_0 * -0.125), Float64(d * Float64(l * sqrt(Float64(l / h))))) / Float64(l * l));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.35e-158], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -8.2e-293], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e-173], N[(N[(N[(1.0 + N[(N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[l, 8.5e+103], N[(N[(N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * N[(t$95$0 * -0.125), $MachinePrecision] + N[(d * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.3499999999999999e-158

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      2. lower-/.f64 55.8

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.3499999999999999e-158 < l < -8.19999999999999975e-293

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{d}}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)}{d}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)} \]

   if -8.19999999999999975e-293 < l < 1.20000000000000008e-173

   1. Initial program 60.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 47.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{h}{d}}}} \]

   6. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right) \cdot d}{\sqrt{h \cdot \ell}}} \]

   if 1.20000000000000008e-173 < l < 8.4999999999999992e103

   1. Initial program 73.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]

   7. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}, \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot \left(\sqrt{h \cdot \ell} \cdot -0.125\right)\right)}{\ell \cdot \ell}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}, \sqrt{h \cdot \ell} \cdot -0.125, d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)\right)}{\color{blue}{\ell} \cdot \ell} \]

   if 8.4999999999999992e103 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 68.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.9%

      \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 22: 64.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(h \cdot \left(D \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right), -0.125 \cdot \left(M \cdot M\right), 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right) \cdot d}{t\_0}\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, \ell \cdot \sqrt{\frac{\ell}{h}}, t\_0 \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)\right)}{\ell}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* h l))))
   (if (<= l -5e-310)
     (/
      (*
       (*
        (sqrt (/ d l))
        (fma (* h (* D (/ D (* l (* d d))))) (* -0.125 (* M M)) 1.0))
       (sqrt (- d)))
      (sqrt (- h)))
     (if (<= l 1.95e-176)
       (/
        (*
         (+ 1.0 (/ (* (* D (* M (* M D))) (* h -0.5)) (* (* (* d d) 4.0) l)))
         d)
        t_0)
       (if (<= l 1.05e+104)
         (/
          (/
           (fma
            d
            (* l (sqrt (/ l h)))
            (* t_0 (* -0.125 (/ (* (* M D) (* M D)) d))))
           l)
          l)
         (* d (/ (/ 1.0 (sqrt h)) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h * l));
	double tmp;
	if (l <= -5e-310) {
		tmp = ((sqrt((d / l)) * fma((h * (D * (D / (l * (d * d))))), (-0.125 * (M * M)), 1.0)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= 1.95e-176) {
		tmp = ((1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l))) * d) / t_0;
	} else if (l <= 1.05e+104) {
		tmp = (fma(d, (l * sqrt((l / h))), (t_0 * (-0.125 * (((M * D) * (M * D)) / d)))) / l) / l;
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h * l))
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(Float64(Float64(sqrt(Float64(d / l)) * fma(Float64(h * Float64(D * Float64(D / Float64(l * Float64(d * d))))), Float64(-0.125 * Float64(M * M)), 1.0)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= 1.95e-176)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(D * Float64(M * Float64(M * D))) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * 4.0) * l))) * d) / t_0);
	elseif (l <= 1.05e+104)
		tmp = Float64(Float64(fma(d, Float64(l * sqrt(Float64(l / h))), Float64(t_0 * Float64(-0.125 * Float64(Float64(Float64(M * D) * Float64(M * D)) / d)))) / l) / l);
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5e-310], N[(N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h * N[(D * N[(D / N[(l * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e-176], N[(N[(N[(1.0 + N[(N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[l, 1.05e+104], N[(N[(N[(d * N[(l * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-0.125 * N[(N[(N[(M * D), $MachinePrecision] * N[(M * D), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / l), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 63.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in M around 0

      \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-1}{8} \cdot \frac{{D}^{2} \cdot \color{blue}{\left(h \cdot {M}^{2}\right)}}{{d}^{2} \cdot \ell}\right)\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-1}{8} \cdot \frac{\color{blue}{\left({D}^{2} \cdot h\right) \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{-1}{8} \cdot \color{blue}{\left(\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot {M}^{2}\right)}\right)\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {M}^{2}}\right)\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {M}^{2} + 1\right)}\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\left(\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \frac{-1}{8}\right)} \cdot {M}^{2} + 1\right)\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell} \cdot \left(\frac{-1}{8} \cdot {M}^{2}\right)} + 1\right)\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(\frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}, \frac{-1}{8} \cdot {M}^{2}, 1\right)}\right) \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

   7. Applied rewrites 58.9%

      \[\leadsto \frac{\left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\mathsf{fma}\left(h \cdot \left(D \cdot \frac{D}{\ell \cdot \left(d \cdot d\right)}\right), -0.125 \cdot \left(M \cdot M\right), 1\right)}\right) \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -4.999999999999985e-310 < l < 1.9499999999999999e-176

   1. Initial program 65.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 48.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{h}{d}}}} \]

   6. Applied rewrites 69.1%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right) \cdot d}{\sqrt{h \cdot \ell}}} \]

   if 1.9499999999999999e-176 < l < 1.0499999999999999e104

   1. Initial program 74.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{h \cdot \ell}\right) + d \cdot \sqrt{\frac{{\ell}^{3}}{h}}}{{\ell}^{2}}} \]

   7. Applied rewrites 59.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(d, \sqrt{\frac{\ell \cdot \left(\ell \cdot \ell\right)}{h}}, \left(D \cdot \left(D \cdot \frac{M \cdot M}{d}\right)\right) \cdot \left(\sqrt{h \cdot \ell} \cdot -0.125\right)\right)}{\ell \cdot \ell}} \]

   9. Applied rewrites 90.0%

      \[\leadsto \frac{\frac{\mathsf{fma}\left(d, \ell \cdot \sqrt{\frac{\ell}{h}}, \sqrt{h \cdot \ell} \cdot \left(-0.125 \cdot \frac{\left(M \cdot D\right) \cdot \left(M \cdot D\right)}{d}\right)\right)}{\ell}}{\color{blue}{\ell}} \]

   if 1.0499999999999999e104 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 68.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.9%

      \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 23: 57.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -8.2 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right) \cdot d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.35e-158)
   (/ (* (sqrt (/ d l)) (sqrt (- d))) (sqrt (- h)))
   (if (<= l -8.2e-293)
     (* (sqrt (/ h (* l (* l l)))) (* (* D (* D (* M M))) (/ 0.125 d)))
     (if (<= l 3.8e+104)
       (/
        (*
         (+ 1.0 (/ (* (* D (* M (* M D))) (* h -0.5)) (* (* (* d d) 4.0) l)))
         d)
        (sqrt (* h l)))
       (* d (/ (/ 1.0 (sqrt h)) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.35e-158) {
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	} else if (l <= -8.2e-293) {
		tmp = sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	} else if (l <= 3.8e+104) {
		tmp = ((1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l))) * d) / sqrt((h * l));
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	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 <= (-1.35d-158)) then
        tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h)
    else if (l <= (-8.2d-293)) then
        tmp = sqrt((h / (l * (l * l)))) * ((d_1 * (d_1 * (m * m))) * (0.125d0 / d))
    else if (l <= 3.8d+104) then
        tmp = ((1.0d0 + (((d_1 * (m * (m * d_1))) * (h * (-0.5d0))) / (((d * d) * 4.0d0) * l))) * d) / sqrt((h * l))
    else
        tmp = d * ((1.0d0 / sqrt(h)) / sqrt(l))
    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 <= -1.35e-158) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt(-d)) / Math.sqrt(-h);
	} else if (l <= -8.2e-293) {
		tmp = Math.sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	} else if (l <= 3.8e+104) {
		tmp = ((1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l))) * d) / Math.sqrt((h * l));
	} else {
		tmp = d * ((1.0 / Math.sqrt(h)) / Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.35e-158:
		tmp = (math.sqrt((d / l)) * math.sqrt(-d)) / math.sqrt(-h)
	elif l <= -8.2e-293:
		tmp = math.sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d))
	elif l <= 3.8e+104:
		tmp = ((1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l))) * d) / math.sqrt((h * l))
	else:
		tmp = d * ((1.0 / math.sqrt(h)) / math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.35e-158)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (l <= -8.2e-293)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(D * Float64(D * Float64(M * M))) * Float64(0.125 / d)));
	elseif (l <= 3.8e+104)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(D * Float64(M * Float64(M * D))) * Float64(h * -0.5)) / Float64(Float64(Float64(d * d) * 4.0) * l))) * d) / sqrt(Float64(h * l)));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.35e-158)
		tmp = (sqrt((d / l)) * sqrt(-d)) / sqrt(-h);
	elseif (l <= -8.2e-293)
		tmp = sqrt((h / (l * (l * l)))) * ((D * (D * (M * M))) * (0.125 / d));
	elseif (l <= 3.8e+104)
		tmp = ((1.0 + (((D * (M * (M * D))) * (h * -0.5)) / (((d * d) * 4.0) * l))) * d) / sqrt((h * l));
	else
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.35e-158], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -8.2e-293], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e+104], N[(N[(N[(1.0 + N[(N[(N[(D * N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * 4.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.3499999999999999e-158

   1. Initial program 66.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 62.2%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right) \cdot \sqrt{-d}}{\sqrt{-h}}} \]

   5. Taylor expanded in d around inf

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \]

      2. lower-/.f64 55.8

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   7. Applied rewrites 55.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-d}}{\sqrt{-h}} \]

   if -1.3499999999999999e-158 < l < -8.19999999999999975e-293

   1. Initial program 61.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in h around -inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \cdot \sqrt{\frac{h}{{\ell}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{h}{{\ell}^{3}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      6. cube-mult N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot \left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{{\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\color{blue}{\ell \cdot {\ell}^{2}}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{d}\right)\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)}{d}}\right)\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{8} \cdot \left({D}^{2} \cdot \left({M}^{2} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)}{d}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{0.125}{d}\right)} \]

   if -8.19999999999999975e-293 < l < 3.79999999999999969e104

   1. Initial program 69.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{\left(d \cdot d\right) \cdot 4}, -0.5 \cdot \frac{h}{\ell}, 1\right)}{\sqrt{\frac{h}{d}}}} \]

   6. Applied rewrites 70.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{\left(D \cdot \left(M \cdot \left(M \cdot D\right)\right)\right) \cdot \left(h \cdot -0.5\right)}{\left(\left(d \cdot d\right) \cdot 4\right) \cdot \ell}\right) \cdot d}{\sqrt{h \cdot \ell}}} \]

   if 3.79999999999999969e104 < l

   1. Initial program 62.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 68.6

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.9%

      \[\leadsto d \cdot \frac{\frac{1}{\sqrt{h}}}{\color{blue}{\sqrt{\ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 24: 46.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 3.4 \cdot 10^{-270}:\\ \;\;\;\;\frac{d}{-\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 3.4e-270) (/ d (- (sqrt (* h l)))) (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 3.4e-270) {
		tmp = d / -sqrt((h * l));
	} 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 (d <= 3.4d-270) then
        tmp = d / -sqrt((h * l))
    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 (d <= 3.4e-270) {
		tmp = d / -Math.sqrt((h * l));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 3.4e-270:
		tmp = d / -math.sqrt((h * l))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 3.4e-270)
		tmp = Float64(d / Float64(-sqrt(Float64(h * l))));
	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 (d <= 3.4e-270)
		tmp = d / -sqrt((h * l));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 3.4e-270], N[(d / (-N[Sqrt[N[(h * l), $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 d < 3.4000000000000001e-270

   1. Initial program 64.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 7.7

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 7.7%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 7.7%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 7.7%

      \[\leadsto \frac{d}{\frac{1}{\color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}}} \]

   10. Taylor expanded in h around -inf

       \[\leadsto \frac{d}{-1 \cdot \color{blue}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 43.3%

       \[\leadsto \frac{d}{-\sqrt{h \cdot \ell}} \]

   if 3.4000000000000001e-270 < d

   1. Initial program 69.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 53.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 53.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 60.0%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 27.0% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((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 / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. lower-*.f64 29.6

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

5. Applied rewrites 29.6%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 29.7%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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)))))