Henrywood and Agarwal, Equation (12)

Percentage Accurate: 67.0% → 85.0%

Time: 16.3s

Alternatives: 22

Speedup: 1.9×

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: 67.0% 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: 85.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\right) \cdot D\\ t_1 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{t\_0}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ t_2 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_3 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\\ t_4 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot t\_2\\ t_5 := t\_3 \cdot t\_4\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{t\_0}{\ell}\right) \cdot t\_4\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 10^{+249}:\\ \;\;\;\;\left({\left(\frac{\ell}{d}\right)}^{-0.5} \cdot t\_2\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M) D))
        (t_1
         (/
          (* (fabs d) (fma (* -0.5 h) (/ (pow t_0 2.0) l) 1.0))
          (sqrt (* l h))))
        (t_2 (pow (/ d h) (/ 1.0 2.0)))
        (t_3
         (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
        (t_4 (* (pow (/ d l) (/ 1.0 2.0)) t_2))
        (t_5 (* t_3 t_4)))
   (if (<= t_5 -1e-64)
     (*
      (- 1.0 (* (/ (* (/ M d) (* (* 0.5 D) 0.5)) (pow h -1.0)) (/ t_0 l)))
      t_4)
     (if (<= t_5 0.0)
       t_1
       (if (<= t_5 1e+249) (* (* (pow (/ l d) -0.5) t_2) t_3) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * M) * D;
	double t_1 = (fabs(d) * fma((-0.5 * h), (pow(t_0, 2.0) / l), 1.0)) / sqrt((l * h));
	double t_2 = pow((d / h), (1.0 / 2.0));
	double t_3 = 1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)));
	double t_4 = pow((d / l), (1.0 / 2.0)) * t_2;
	double t_5 = t_3 * t_4;
	double tmp;
	if (t_5 <= -1e-64) {
		tmp = (1.0 - ((((M / d) * ((0.5 * D) * 0.5)) / pow(h, -1.0)) * (t_0 / l))) * t_4;
	} else if (t_5 <= 0.0) {
		tmp = t_1;
	} else if (t_5 <= 1e+249) {
		tmp = (pow((l / d), -0.5) * t_2) * t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * M) * D)
	t_1 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((t_0 ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	t_2 = Float64(d / h) ^ Float64(1.0 / 2.0)
	t_3 = Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0))))
	t_4 = Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * t_2)
	t_5 = Float64(t_3 * t_4)
	tmp = 0.0
	if (t_5 <= -1e-64)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(M / d) * Float64(Float64(0.5 * D) * 0.5)) / (h ^ -1.0)) * Float64(t_0 / l))) * t_4);
	elseif (t_5 <= 0.0)
		tmp = t_1;
	elseif (t_5 <= 1e+249)
		tmp = Float64(Float64((Float64(l / d) ^ -0.5) * t_2) * t_3);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-64], N[(N[(1.0 - N[(N[(N[(N[(M / d), $MachinePrecision] * N[(N[(0.5 * D), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 0.0], t$95$1, If[LessEqual[t$95$5, 1e+249], N[(N[(N[Power[N[(l / d), $MachinePrecision], -0.5], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $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)))) < -9.99999999999999965e-65

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 94.4%

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

   if -9.99999999999999965e-65 < (*.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 9.9999999999999992e248 < (*.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 19.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.4%

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 72.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(1 - \frac{\frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right)}{{h}^{-1}} \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left({\left(\frac{\ell}{d}\right)}^{-0.5} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\\ t_1 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(t\_0 \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ t_3 := 1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\\ t_4 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot t\_1\\ t_5 := t\_3 \cdot t\_4\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{\ell} \cdot t\_0\right) \cdot \left(-h\right), \frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right), 1\right) \cdot t\_4\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 10^{+249}:\\ \;\;\;\;\left({\left(\frac{\ell}{d}\right)}^{-0.5} \cdot t\_1\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) M))
        (t_1 (pow (/ d h) (/ 1.0 2.0)))
        (t_2
         (/
          (* (fabs d) (fma (* -0.5 h) (/ (pow (* t_0 D) 2.0) l) 1.0))
          (sqrt (* l h))))
        (t_3
         (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
        (t_4 (* (pow (/ d l) (/ 1.0 2.0)) t_1))
        (t_5 (* t_3 t_4)))
   (if (<= t_5 -1e-64)
     (* (fma (* (* (/ D l) t_0) (- h)) (* (/ M d) (* (* 0.5 D) 0.5)) 1.0) t_4)
     (if (<= t_5 0.0)
       t_2
       (if (<= t_5 1e+249) (* (* (pow (/ l d) -0.5) t_1) t_3) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 / d) * M;
	double t_1 = pow((d / h), (1.0 / 2.0));
	double t_2 = (fabs(d) * fma((-0.5 * h), (pow((t_0 * D), 2.0) / l), 1.0)) / sqrt((l * h));
	double t_3 = 1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)));
	double t_4 = pow((d / l), (1.0 / 2.0)) * t_1;
	double t_5 = t_3 * t_4;
	double tmp;
	if (t_5 <= -1e-64) {
		tmp = fma((((D / l) * t_0) * -h), ((M / d) * ((0.5 * D) * 0.5)), 1.0) * t_4;
	} else if (t_5 <= 0.0) {
		tmp = t_2;
	} else if (t_5 <= 1e+249) {
		tmp = (pow((l / d), -0.5) * t_1) * t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 / d) * M)
	t_1 = Float64(d / h) ^ Float64(1.0 / 2.0)
	t_2 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((Float64(t_0 * D) ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	t_3 = Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0))))
	t_4 = Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * t_1)
	t_5 = Float64(t_3 * t_4)
	tmp = 0.0
	if (t_5 <= -1e-64)
		tmp = Float64(fma(Float64(Float64(Float64(D / l) * t_0) * Float64(-h)), Float64(Float64(M / d) * Float64(Float64(0.5 * D) * 0.5)), 1.0) * t_4);
	elseif (t_5 <= 0.0)
		tmp = t_2;
	elseif (t_5 <= 1e+249)
		tmp = Float64(Float64((Float64(l / d) ^ -0.5) * t_1) * t_3);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[N[(t$95$0 * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-64], N[(N[(N[(N[(N[(D / l), $MachinePrecision] * t$95$0), $MachinePrecision] * (-h)), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(0.5 * D), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 0.0], t$95$2, If[LessEqual[t$95$5, 1e+249], N[(N[(N[Power[N[(l / d), $MachinePrecision], -0.5], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $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)))) < -9.99999999999999965e-65

   1. Initial program 87.0%

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

   4. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 94.4

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if -9.99999999999999965e-65 < (*.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 9.9999999999999992e248 < (*.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 19.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.4%

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 72.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{\ell} \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \left(-h\right), \frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right), 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left({\left(\frac{\ell}{d}\right)}^{-0.5} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{d} \cdot M\\ t_1 := {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_2 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot t\_1\\ t_3 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(t\_0 \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{\ell} \cdot t\_0\right) \cdot \left(-h\right), \frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right), 1\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ 0.5 d) M))
        (t_1 (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        (t_2
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          t_1))
        (t_3
         (/
          (* (fabs d) (fma (* -0.5 h) (/ (pow (* t_0 D) 2.0) l) 1.0))
          (sqrt (* l h)))))
   (if (<= t_2 -1e-64)
     (* (fma (* (* (/ D l) t_0) (- h)) (* (/ M d) (* (* 0.5 D) 0.5)) 1.0) t_1)
     (if (<= t_2 0.0)
       t_3
       (if (<= t_2 1e+249)
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (fma (* -0.125 (pow (/ (/ d D) M) -2.0)) (/ h l) 1.0)))
         t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (0.5 / d) * M;
	double t_1 = pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0));
	double t_2 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * t_1;
	double t_3 = (fabs(d) * fma((-0.5 * h), (pow((t_0 * D), 2.0) / l), 1.0)) / sqrt((l * h));
	double tmp;
	if (t_2 <= -1e-64) {
		tmp = fma((((D / l) * t_0) * -h), ((M / d) * ((0.5 * D) * 0.5)), 1.0) * t_1;
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= 1e+249) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * fma((-0.125 * pow(((d / D) / M), -2.0)), (h / l), 1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(0.5 / d) * M)
	t_1 = Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))
	t_2 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * t_1)
	t_3 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((Float64(t_0 * D) ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_2 <= -1e-64)
		tmp = Float64(fma(Float64(Float64(Float64(D / l) * t_0) * Float64(-h)), Float64(Float64(M / d) * Float64(Float64(0.5 * D) * 0.5)), 1.0) * t_1);
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= 1e+249)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * fma(Float64(-0.125 * (Float64(Float64(d / D) / M) ^ -2.0)), Float64(h / l), 1.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[N[(t$95$0 * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-64], N[(N[(N[(N[(N[(D / l), $MachinePrecision] * t$95$0), $MachinePrecision] * (-h)), $MachinePrecision] * N[(N[(M / d), $MachinePrecision] * N[(N[(0.5 * D), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, 1e+249], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $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)))) < -9.99999999999999965e-65

   1. Initial program 87.0%

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

   4. Applied rewrites 92.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-*.f64 94.4

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 93.3

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

   6. Applied rewrites 93.3%

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

   if -9.99999999999999965e-65 < (*.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 9.9999999999999992e248 < (*.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 19.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.4%

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 72.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{D}{\ell} \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \left(-h\right), \frac{M}{d} \cdot \left(\left(0.5 \cdot D\right) \cdot 0.5\right), 1\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\right) \cdot D\\ t_1 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{t\_0}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ t_2 := {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ t_3 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot t\_2\right)\\ t_4 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(1 - \left(\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot h\right) \cdot 0.5\right) \cdot \frac{t\_0}{\ell}\right) \cdot \left(t\_4 \cdot t\_2\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(t\_4 \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M) D))
        (t_1
         (/
          (* (fabs d) (fma (* -0.5 h) (/ (pow t_0 2.0) l) 1.0))
          (sqrt (* l h))))
        (t_2 (pow (/ d h) (/ 1.0 2.0)))
        (t_3
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) t_2)))
        (t_4 (sqrt (/ d l))))
   (if (<= t_3 -1e-64)
     (* (- 1.0 (* (* (* (* (* (/ 0.5 d) D) M) h) 0.5) (/ t_0 l))) (* t_4 t_2))
     (if (<= t_3 0.0)
       t_1
       (if (<= t_3 1e+249)
         (*
          (sqrt (/ d h))
          (* t_4 (fma (* -0.125 (pow (/ (/ d D) M) -2.0)) (/ h l) 1.0)))
         t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * M) * D;
	double t_1 = (fabs(d) * fma((-0.5 * h), (pow(t_0, 2.0) / l), 1.0)) / sqrt((l * h));
	double t_2 = pow((d / h), (1.0 / 2.0));
	double t_3 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * t_2);
	double t_4 = sqrt((d / l));
	double tmp;
	if (t_3 <= -1e-64) {
		tmp = (1.0 - ((((((0.5 / d) * D) * M) * h) * 0.5) * (t_0 / l))) * (t_4 * t_2);
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 1e+249) {
		tmp = sqrt((d / h)) * (t_4 * fma((-0.125 * pow(((d / D) / M), -2.0)), (h / l), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * M) * D)
	t_1 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((t_0 ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	t_2 = Float64(d / h) ^ Float64(1.0 / 2.0)
	t_3 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * t_2))
	t_4 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_3 <= -1e-64)
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(0.5 / d) * D) * M) * h) * 0.5) * Float64(t_0 / l))) * Float64(t_4 * t_2));
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 1e+249)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(t_4 * fma(Float64(-0.125 * (Float64(Float64(d / D) / M) ^ -2.0)), Float64(h / l), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -1e-64], N[(N[(1.0 - N[(N[(N[(N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision] * h), $MachinePrecision] * 0.5), $MachinePrecision] * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 1e+249], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(t$95$4 * N[(N[(-0.125 * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $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)))) < -9.99999999999999965e-65

   1. Initial program 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 94.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/l* N/A

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

      14. associate-*l/ N/A

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

      15. lift-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 91.2

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 91.2

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

   8. Applied rewrites 91.2%

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

   if -9.99999999999999965e-65 < (*.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 9.9999999999999992e248 < (*.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 19.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.4%

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 72.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(1 - \left(\left(\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right) \cdot h\right) \cdot 0.5\right) \cdot \frac{\left(\frac{0.5}{d} \cdot M\right) \cdot D}{\ell}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot t\_1}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d h)))
        (t_2
         (/
          (*
           (fabs d)
           (fma (* -0.5 h) (/ (pow (* (* (/ 0.5 d) M) D) 2.0) l) 1.0))
          (sqrt (* l h)))))
   (if (<= t_0 -1e+17)
     (/
      (* (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0) t_1)
      (sqrt (/ l d)))
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 1e+249)
         (*
          t_1
          (*
           (sqrt (/ d l))
           (fma (* -0.125 (pow (/ (/ d D) M) -2.0)) (/ h l) 1.0)))
         t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / h));
	double t_2 = (fabs(d) * fma((-0.5 * h), (pow((((0.5 / d) * M) * D), 2.0) / l), 1.0)) / sqrt((l * h));
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = (fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0) * t_1) / sqrt((l / d));
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+249) {
		tmp = t_1 * (sqrt((d / l)) * fma((-0.125 * pow(((d / D) / M), -2.0)), (h / l), 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / h))
	t_2 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((Float64(Float64(Float64(0.5 / d) * M) * D) ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= -1e+17)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0) * t_1) / sqrt(Float64(l / d)));
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+249)
		tmp = Float64(t_1 * Float64(sqrt(Float64(d / l)) * fma(Float64(-0.125 * (Float64(Float64(d / D) / M) ^ -2.0)), Float64(h / l), 1.0)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+17], N[(N[(N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+249], N[(t$95$1 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.125 * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $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)))) < -1e17

   1. Initial program 86.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 32.2%

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

   8. Applied rewrites 86.6%

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

   if -1e17 < (*.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 9.9999999999999992e248 < (*.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 20.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 20.8

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

   4. Applied rewrites 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.3%

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

   8. Applied rewrites 69.9%

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

   10. Applied rewrites 73.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{h}}}{\sqrt{\frac{\ell}{d}}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;t\_2 \cdot \left(t\_1 \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3
         (/
          (*
           (fabs d)
           (fma (* -0.5 h) (/ (pow (* (* (/ 0.5 d) M) D) 2.0) l) 1.0))
          (sqrt (* l h)))))
   (if (<= t_0 -1e-64)
     (* (* (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0) t_1) t_2)
     (if (<= t_0 0.0)
       t_3
       (if (<= t_0 1e+249)
         (* t_2 (* t_1 (fma (* -0.125 (pow (/ (/ d D) M) -2.0)) (/ h l) 1.0)))
         t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = (fabs(d) * fma((-0.5 * h), (pow((((0.5 / d) * M) * D), 2.0) / l), 1.0)) / sqrt((l * h));
	double tmp;
	if (t_0 <= -1e-64) {
		tmp = (fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = t_3;
	} else if (t_0 <= 1e+249) {
		tmp = t_2 * (t_1 * fma((-0.125 * pow(((d / D) / M), -2.0)), (h / l), 1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((Float64(Float64(Float64(0.5 / d) * M) * D) ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= -1e-64)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = t_3;
	elseif (t_0 <= 1e+249)
		tmp = Float64(t_2 * Float64(t_1 * fma(Float64(-0.125 * (Float64(Float64(d / D) / M) ^ -2.0)), Float64(h / l), 1.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-64], N[(N[(N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$3, If[LessEqual[t$95$0, 1e+249], N[(t$95$2 * N[(t$95$1 * N[(N[(-0.125 * N[Power[N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision] + 1.0), $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)))) < -9.99999999999999965e-65

   1. Initial program 87.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.0

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

   4. Applied rewrites 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

   8. Applied rewrites 86.8%

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

   if -9.99999999999999965e-65 < (*.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 9.9999999999999992e248 < (*.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 19.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.4%

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 72.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left({\left(\frac{\frac{d}{D}}{M}\right)}^{-2} \cdot -0.125, \frac{h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(-0.125 \cdot {\left(\frac{\frac{d}{D}}{M}\right)}^{-2}, \frac{h}{\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot t\_1\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h)))
        (t_3
         (/
          (*
           (fabs d)
           (fma (* -0.5 h) (/ (pow (* (* (/ 0.5 d) M) D) 2.0) l) 1.0))
          (sqrt (* l h)))))
   (if (<= t_0 -1e-64)
     (* (* (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0) t_1) t_2)
     (if (<= t_0 0.0) t_3 (if (<= t_0 1e+249) (* (* 1.0 t_1) t_2) t_3)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double t_3 = (fabs(d) * fma((-0.5 * h), (pow((((0.5 / d) * M) * D), 2.0) / l), 1.0)) / sqrt((l * h));
	double tmp;
	if (t_0 <= -1e-64) {
		tmp = (fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0) * t_1) * t_2;
	} else if (t_0 <= 0.0) {
		tmp = t_3;
	} else if (t_0 <= 1e+249) {
		tmp = (1.0 * t_1) * t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	t_3 = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((Float64(Float64(Float64(0.5 / d) * M) * D) ^ 2.0) / l), 1.0)) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= -1e-64)
		tmp = Float64(Float64(fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0) * t_1) * t_2);
	elseif (t_0 <= 0.0)
		tmp = t_3;
	elseif (t_0 <= 1e+249)
		tmp = Float64(Float64(1.0 * t_1) * t_2);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-64], N[(N[(N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$3, If[LessEqual[t$95$0, 1e+249], N[(N[(1.0 * t$95$1), $MachinePrecision] * 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)))) < -9.99999999999999965e-65

   1. Initial program 87.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.0

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

   4. Applied rewrites 87.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

   8. Applied rewrites 86.8%

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

   if -9.99999999999999965e-65 < (*.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 9.9999999999999992e248 < (*.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 19.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 13.4%

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 72.9%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{-64}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := \mathsf{fma}\left(\frac{\frac{D}{d} \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{\ell}, h, 1\right) \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+97}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_1 \cdot 1\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2
         (* (fma (* (/ (* (/ D d) D) d) (/ (* (* M M) -0.125) l)) h 1.0) t_1)))
   (if (<= t_0 -5e+97)
     t_2
     (if (<= t_0 0.0)
       (* t_1 1.0)
       (if (<= t_0 1e+249) (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h))) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = fma(((((D / d) * D) / d) * (((M * M) * -0.125) / l)), h, 1.0) * t_1;
	double tmp;
	if (t_0 <= -5e+97) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = t_1 * 1.0;
	} else if (t_0 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(fma(Float64(Float64(Float64(Float64(D / d) * D) / d) * Float64(Float64(Float64(M * M) * -0.125) / l)), h, 1.0) * t_1)
	tmp = 0.0
	if (t_0 <= -5e+97)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(t_1 * 1.0);
	elseif (t_0 <= 1e+249)
		tmp = Float64(Float64(1.0 * 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[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+97], t$95$2, If[LessEqual[t$95$0, 0.0], N[(t$95$1 * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $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)))) < -4.99999999999999999e97 or 9.9999999999999992e248 < (*.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 54.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 24.5%

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. times-frac N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. associate-/r* N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. associate-/l* N/A

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

       16. lower-*.f64 N/A

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

       17. lower-/.f64 65.0

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

   11. Applied rewrites 65.0%

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

   if -4.99999999999999999e97 < (*.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 43.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 19.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. lower-/.f64 N/A

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

      21. rem-sqrt-square N/A

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

      22. lower-fabs.f64 N/A

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

      23. lower-sqrt.f64 66.0

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

   9. Applied rewrites 66.0%

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

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -5 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{D}{d} \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{D}{d} \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ t_2 := t\_1 \cdot 1\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\left(\frac{\frac{M}{d} \cdot M}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ (fabs d) (sqrt (* l h))))
        (t_2 (* t_1 1.0)))
   (if (<= t_0 -2e-131)
     (* (* (* (/ (* (/ M d) M) d) h) (/ (* (* D D) -0.125) l)) t_1)
     (if (<= t_0 0.0)
       t_2
       (if (<= t_0 1e+249) (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h))) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = fabs(d) / sqrt((l * h));
	double t_2 = t_1 * 1.0;
	double tmp;
	if (t_0 <= -2e-131) {
		tmp = (((((M / d) * M) / d) * h) * (((D * D) * -0.125) / l)) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+249) {
		tmp = (1.0 * 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 = (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_1 = abs(d) / sqrt((l * h))
    t_2 = t_1 * 1.0d0
    if (t_0 <= (-2d-131)) then
        tmp = (((((m / d) * m) / d) * h) * (((d_1 * d_1) * (-0.125d0)) / l)) * t_1
    else if (t_0 <= 0.0d0) then
        tmp = t_2
    else if (t_0 <= 1d+249) then
        tmp = (1.0d0 * 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 = (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_1 = Math.abs(d) / Math.sqrt((l * h));
	double t_2 = t_1 * 1.0;
	double tmp;
	if (t_0 <= -2e-131) {
		tmp = (((((M / d) * M) / d) * h) * (((D * D) * -0.125) / l)) * t_1;
	} else if (t_0 <= 0.0) {
		tmp = t_2;
	} else if (t_0 <= 1e+249) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_1 = math.fabs(d) / math.sqrt((l * h))
	t_2 = t_1 * 1.0
	tmp = 0
	if t_0 <= -2e-131:
		tmp = (((((M / d) * M) / d) * h) * (((D * D) * -0.125) / l)) * t_1
	elif t_0 <= 0.0:
		tmp = t_2
	elif t_0 <= 1e+249:
		tmp = (1.0 * 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(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	t_2 = Float64(t_1 * 1.0)
	tmp = 0.0
	if (t_0 <= -2e-131)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M / d) * M) / d) * h) * Float64(Float64(Float64(D * D) * -0.125) / l)) * t_1);
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+249)
		tmp = Float64(Float64(1.0 * 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 = (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_1 = abs(d) / sqrt((l * h));
	t_2 = t_1 * 1.0;
	tmp = 0.0;
	if (t_0 <= -2e-131)
		tmp = (((((M / d) * M) / d) * h) * (((D * D) * -0.125) / l)) * t_1;
	elseif (t_0 <= 0.0)
		tmp = t_2;
	elseif (t_0 <= 1e+249)
		tmp = (1.0 * 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[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-131], N[(N[(N[(N[(N[(N[(M / d), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D * D), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 0.0], t$95$2, If[LessEqual[t$95$0, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $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)))) < -2e-131

   1. Initial program 87.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-undiv N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. metadata-eval N/A

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

      19. pow-flip N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 31.5%

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

   8. Applied rewrites 82.4%

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

   9. Taylor expanded in d around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. associate-*l/ N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. associate-/r* N/A

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

       14. lower-/.f64 N/A

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

       15. unpow2 N/A

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

       16. associate-/l* N/A

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

       17. lower-*.f64 N/A

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

       18. lower-/.f64 66.8

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

   11. Applied rewrites 66.8%

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

   if -2e-131 < (*.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 9.9999999999999992e248 < (*.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 19.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.0

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

   4. Applied rewrites 19.0%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 28.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. lower-/.f64 N/A

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

      21. rem-sqrt-square N/A

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

      22. lower-fabs.f64 N/A

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

      23. lower-sqrt.f64 62.1

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

   9. Applied rewrites 62.1%

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

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

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

   3. Taylor expanded in d around inf

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\left(\frac{\frac{M}{d} \cdot M}{d} \cdot h\right) \cdot \frac{\left(D \cdot D\right) \cdot -0.125}{\ell}\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(-t\_0\right)\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_2 (* (/ (fabs d) (sqrt (* l h))) 1.0)))
   (if (<= t_1 -2e-131)
     (* (* (/ 1.0 (sqrt (/ l d))) (- t_0)) 1.0)
     (if (<= t_1 0.0)
       t_2
       (if (<= t_1 1e+249) (* (* 1.0 (sqrt (/ d l))) t_0) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_2 = (fabs(d) / sqrt((l * h))) * 1.0;
	double tmp;
	if (t_1 <= -2e-131) {
		tmp = ((1.0 / sqrt((l / d))) * -t_0) * 1.0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * t_0;
	} 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 = sqrt((d / h))
    t_1 = (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    t_2 = (abs(d) / sqrt((l * h))) * 1.0d0
    if (t_1 <= (-2d-131)) then
        tmp = ((1.0d0 / sqrt((l / d))) * -t_0) * 1.0d0
    else if (t_1 <= 0.0d0) then
        tmp = t_2
    else if (t_1 <= 1d+249) then
        tmp = (1.0d0 * sqrt((d / l))) * t_0
    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.sqrt((d / h));
	double t_1 = (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double t_2 = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	double tmp;
	if (t_1 <= -2e-131) {
		tmp = ((1.0 / Math.sqrt((l / d))) * -t_0) * 1.0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1e+249) {
		tmp = (1.0 * Math.sqrt((d / l))) * t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / h))
	t_1 = (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	t_2 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	tmp = 0
	if t_1 <= -2e-131:
		tmp = ((1.0 / math.sqrt((l / d))) * -t_0) * 1.0
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= 1e+249:
		tmp = (1.0 * math.sqrt((d / l))) * t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_2 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	tmp = 0.0
	if (t_1 <= -2e-131)
		tmp = Float64(Float64(Float64(1.0 / sqrt(Float64(l / d))) * Float64(-t_0)) * 1.0);
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1e+249)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * t_0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / h));
	t_1 = (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	t_2 = (abs(d) / sqrt((l * h))) * 1.0;
	tmp = 0.0;
	if (t_1 <= -2e-131)
		tmp = ((1.0 / sqrt((l / d))) * -t_0) * 1.0;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1e+249)
		tmp = (1.0 * sqrt((d / l))) * t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-131], N[(N[(N[(1.0 / N[Sqrt[N[(l / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-t$95$0)), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $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)))) < -2e-131

   1. Initial program 87.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 0.9%

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

   8. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 16.6

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

   10. Applied rewrites 16.6%

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

   if -2e-131 < (*.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 9.9999999999999992e248 < (*.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 19.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 19.0

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

   4. Applied rewrites 19.0%

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

   5. Taylor expanded in d around inf

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

   7. Applied rewrites 28.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow1/2 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lift-sqrt.f64 N/A

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

      9. sqrt-div N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. sqrt-unprod N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-div N/A

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

      20. lower-/.f64 N/A

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

      21. rem-sqrt-square N/A

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

      22. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      23. lower-sqrt.f64 62.1

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   9. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\frac{\ell}{d}}} \cdot \left(-\sqrt{\frac{d}{h}}\right)\right) \cdot 1\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -2e-131)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= t_1 0.0)
       t_0
       (if (<= t_1 1e+249) (* (* 1.0 (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((l * h))) * 1.0;
	double t_1 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-131) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = (abs(d) / sqrt((l * h))) * 1.0d0
    t_1 = (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-2d-131)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (t_1 <= 0.0d0) then
        tmp = t_0
    else if (t_1 <= 1d+249) then
        tmp = (1.0d0 * sqrt((d / l))) * sqrt((d / h))
    else
        tmp = 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.abs(d) / Math.sqrt((l * h))) * 1.0;
	double t_1 = (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-131) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+249) {
		tmp = (1.0 * Math.sqrt((d / l))) * Math.sqrt((d / h));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	t_1 = (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -2e-131:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 1e+249:
		tmp = (1.0 * math.sqrt((d / l))) * math.sqrt((d / h))
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -2e-131)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+249)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	t_1 = (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -2e-131)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+249)
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-131], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $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)))) < -2e-131

   1. Initial program 87.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 0.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 5.9

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   10. Step-by-step derivation

   11. Applied rewrites 8.6%

       \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -2e-131 < (*.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 9.9999999999999992e248 < (*.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 19.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 19.0

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 19.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 28.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      5. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot 1 \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot 1 \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      20. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      21. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      22. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      23. lower-sqrt.f64 62.1

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   9. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-163}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ (fabs d) (sqrt (* l h))) 1.0))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 -2e-131)
     (* (- (sqrt (/ 1.0 (* l h)))) d)
     (if (<= t_1 5e-163)
       t_0
       (if (<= t_1 1e+136) (* (sqrt (* (/ d l) (/ d h))) 1.0) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (fabs(d) / sqrt((l * h))) * 1.0;
	double t_1 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-131) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else if (t_1 <= 5e-163) {
		tmp = t_0;
	} else if (t_1 <= 1e+136) {
		tmp = sqrt(((d / l) * (d / h))) * 1.0;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_0 = (abs(d) / sqrt((l * h))) * 1.0d0
    t_1 = (1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))
    if (t_1 <= (-2d-131)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else if (t_1 <= 5d-163) then
        tmp = t_0
    else if (t_1 <= 1d+136) then
        tmp = sqrt(((d / l) * (d / h))) * 1.0d0
    else
        tmp = 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.abs(d) / Math.sqrt((l * h))) * 1.0;
	double t_1 = (1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= -2e-131) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else if (t_1 <= 5e-163) {
		tmp = t_0;
	} else if (t_1 <= 1e+136) {
		tmp = Math.sqrt(((d / l) * (d / h))) * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	t_1 = (1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_1 <= -2e-131:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	elif t_1 <= 5e-163:
		tmp = t_0
	elif t_1 <= 1e+136:
		tmp = math.sqrt(((d / l) * (d / h))) * 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0)
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= -2e-131)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	elseif (t_1 <= 5e-163)
		tmp = t_0;
	elseif (t_1 <= 1e+136)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (abs(d) / sqrt((l * h))) * 1.0;
	t_1 = (1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_1 <= -2e-131)
		tmp = -sqrt((1.0 / (l * h))) * d;
	elseif (t_1 <= 5e-163)
		tmp = t_0;
	elseif (t_1 <= 1e+136)
		tmp = sqrt(((d / l) * (d / h))) * 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-131], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], If[LessEqual[t$95$1, 5e-163], t$95$0, If[LessEqual[t$95$1, 1e+136], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $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)))) < -2e-131

   1. Initial program 87.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 0.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 5.9

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   10. Step-by-step derivation

   11. Applied rewrites 8.6%

       \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -2e-131 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999977e-163 or 1.00000000000000006e136 < (*.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 32.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 32.5

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 32.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 40.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      5. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot 1 \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot 1 \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      20. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      21. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      22. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      23. lower-sqrt.f64 63.3

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   9. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   if 4.99999999999999977e-163 < (*.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)))) < 1.00000000000000006e136

   1. Initial program 99.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot 1 \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot 1 \]

      7. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}}\right) \cdot 1 \]

      8. pow-prod-down N/A

         \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot 1 \]

      9. unpow1/2 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot 1 \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot 1 \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

      12. lower-*.f64 98.9

          \[\leadsto \sqrt{\color{blue}{\frac{d}{\ell} \cdot \frac{d}{h}}} \cdot 1 \]

   7. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+136}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{t\_0} \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_1 0.0)
     (*
      (/ (fabs d) t_0)
      (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0))
     (if (<= t_1 1e+249)
       (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
       (/
        (* (fabs d) (fma (* -0.5 h) (/ (pow (* (* (/ 0.5 d) M) D) 2.0) l) 1.0))
        t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (fabs(d) / t_0) * fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0);
	} else if (t_1 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = (fabs(d) * fma((-0.5 * h), (pow((((0.5 / d) * M) * D), 2.0) / l), 1.0)) / t_0;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(abs(d) / t_0) * fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0));
	elseif (t_1 <= 1e+249)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(abs(d) * fma(Float64(-0.5 * h), Float64((Float64(Float64(Float64(0.5 / d) * M) * D) ^ 2.0) / l), 1.0)) / t_0);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] * N[(N[(-0.5 * h), $MachinePrecision] * N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 78.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 78.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 78.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 9.9999999999999992e248 < (*.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 17.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 17.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 17.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   10. Applied rewrites 67.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(h \cdot -0.5, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right) \cdot \left|d\right|}{\sqrt{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right| \cdot \mathsf{fma}\left(-0.5 \cdot h, \frac{{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}^{2}}{\ell}, 1\right)}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \left(\frac{0.5}{d} \cdot M\right) \cdot D\\ t_2 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot t\_1\right) \cdot t\_1}{\ell}, h, 1\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (* (* (/ 0.5 d) M) D))
        (t_2 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_0 0.0)
     (* t_2 (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0))
     (if (<= t_0 1e+249)
       (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
       (* (fma (/ (* (* -0.5 t_1) t_1) l) h 1.0) t_2)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = ((0.5 / d) * M) * D;
	double t_2 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_2 * fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0);
	} else if (t_0 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fma((((-0.5 * t_1) * t_1) / l), h, 1.0) * t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(Float64(Float64(0.5 / d) * M) * D)
	t_2 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(t_2 * fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0));
	elseif (t_0 <= 1e+249)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(-0.5 * t_1) * t_1) / l), h, 1.0) * t_2);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(t$95$2 * N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$2), $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 78.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 78.3

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 78.3%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 28.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 9.9999999999999992e248 < (*.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 17.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 17.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 17.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2} \cdot {\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}^{2}}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}^{2} \cdot \frac{-1}{2}}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}^{2}} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(M \cdot \color{blue}{\left(D \cdot \frac{\frac{1}{2}}{d}\right)}\right)}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{\frac{1}{2}}{d}\right)}^{2}\right)} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\color{blue}{\left(\frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. unpow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{2}} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. div-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 65.6%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot -0.5\right)}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.5}{d} \cdot M\right) \cdot D\\ t_1 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_2 := \mathsf{fma}\left(\frac{\left(-0.5 \cdot t\_0\right) \cdot t\_0}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (* (/ 0.5 d) M) D))
        (t_1
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_2
         (*
          (fma (/ (* (* -0.5 t_0) t_0) l) h 1.0)
          (/ (fabs d) (sqrt (* l h))))))
   (if (<= t_1 0.0)
     t_2
     (if (<= t_1 1e+249) (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h))) t_2))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = ((0.5 / d) * M) * D;
	double t_1 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_2 = fma((((-0.5 * t_0) * t_0) / l), h, 1.0) * (fabs(d) / sqrt((l * h)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(Float64(0.5 / d) * M) * D)
	t_1 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_2 = Float64(fma(Float64(Float64(Float64(-0.5 * t_0) * t_0) / l), h, 1.0) * Float64(abs(d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= 1e+249)
		tmp = Float64(Float64(1.0 * 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[(N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision] * D), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(-0.5 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 9.9999999999999992e248 < (*.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 53.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 53.4

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{2} \cdot {\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}^{2}}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}^{2} \cdot \frac{-1}{2}}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}^{2}} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(M \cdot \left(D \cdot \frac{\frac{1}{2}}{d}\right)\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(M \cdot \color{blue}{\left(D \cdot \frac{\frac{1}{2}}{d}\right)}\right)}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{\frac{1}{2}}{d}\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{\left(\color{blue}{\left(M \cdot D\right)} \cdot \frac{\frac{1}{2}}{d}\right)}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{\frac{1}{2}}{d}\right)}^{2}\right)} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\color{blue}{\left(\frac{\frac{1}{2}}{d}\right)}}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{\color{blue}{\frac{1}{2}}}{d}\right)}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\color{blue}{\left(\frac{1}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left({\left(M \cdot D\right)}^{2} \cdot {\left(\frac{1}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. unpow-prod-down N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{2}} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. div-inv N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot \frac{-1}{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{-1}{2}\right)}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 76.9%

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot \left(\left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right) \cdot -0.5\right)}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   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)))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-0.5 \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)\right) \cdot \left(\left(\frac{0.5}{d} \cdot M\right) \cdot D\right)}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ t_1 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{D}{d} \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{\ell}, h, 1\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0)))))
        (t_1 (/ (fabs d) (sqrt (* l h)))))
   (if (<= t_0 0.0)
     (*
      t_1
      (fma (* (* 0.25 (* (/ M d) D)) (* (/ 0.5 d) M)) (* (/ (- h) l) D) 1.0))
     (if (<= t_0 1e+249)
       (* (* 1.0 (sqrt (/ d l))) (sqrt (/ d h)))
       (* (fma (* (/ (* (/ D d) D) d) (/ (* (* M M) -0.125) l)) h 1.0) t_1)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double t_1 = fabs(d) / sqrt((l * h));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1 * fma(((0.25 * ((M / d) * D)) * ((0.5 / d) * M)), ((-h / l) * D), 1.0);
	} else if (t_0 <= 1e+249) {
		tmp = (1.0 * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fma(((((D / d) * D) / d) * (((M * M) * -0.125) / l)), h, 1.0) * t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	t_1 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(t_1 * fma(Float64(Float64(0.25 * Float64(Float64(M / d) * D)) * Float64(Float64(0.5 / d) * M)), Float64(Float64(Float64(-h) / l) * D), 1.0));
	elseif (t_0 <= 1e+249)
		tmp = Float64(Float64(1.0 * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D / d) * D) / d) * Float64(Float64(Float64(M * M) * -0.125) / l)), h, 1.0) * t_1);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(t$95$1 * N[(N[(N[(0.25 * N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[(N[((-h) / l), $MachinePrecision] * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+249], N[(N[(1.0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(D / d), $MachinePrecision] * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] * -0.125), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision] * t$95$1), $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 78.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}} + 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}} + 1\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\frac{\ell}{h}}} + 1\right) \]

      9. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot h} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell}, h, 1\right)} \]

   4. Applied rewrites 78.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)} \]

   6. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right), \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right) + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \color{blue}{\left(\left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   8. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{M}{d} \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   7. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]

   if 9.9999999999999992e248 < (*.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 17.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 17.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 17.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]

   9. Taylor expanded in d around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({D}^{2} \cdot {M}^{2}\right)}{{d}^{2} \cdot \ell}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left({M}^{2} \cdot {D}^{2}\right)}}{{d}^{2} \cdot \ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot {D}^{2}}}{{d}^{2} \cdot \ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(\frac{-1}{8} \cdot {M}^{2}\right) \cdot {D}^{2}}{\color{blue}{\ell \cdot {d}^{2}}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       5. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       7. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{8} \cdot {M}^{2}}{\ell}} \cdot \frac{{D}^{2}}{{d}^{2}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{-1}{8} \cdot {M}^{2}}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       9. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(M \cdot M\right)}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \color{blue}{\left(M \cdot M\right)}}{\ell} \cdot \frac{{D}^{2}}{{d}^{2}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       11. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{{D}^{2}}{\color{blue}{d \cdot d}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       12. associate-/r* N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{\ell} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       13. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{\ell} \cdot \color{blue}{\frac{\frac{{D}^{2}}{d}}{d}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{\frac{\color{blue}{D \cdot D}}{d}}{d}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       15. associate-/l* N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{\color{blue}{D \cdot \frac{D}{d}}}{d}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       16. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{8} \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{\color{blue}{D \cdot \frac{D}{d}}}{d}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

       17. lower-/.f64 58.5

           \[\leadsto \mathsf{fma}\left(\frac{-0.125 \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{D \cdot \color{blue}{\frac{D}{d}}}{d}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   11. Applied rewrites 58.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-0.125 \cdot \left(M \cdot M\right)}{\ell} \cdot \frac{D \cdot \frac{D}{d}}{d}}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 0:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{+249}:\\ \;\;\;\;\left(1 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{D}{d} \cdot D}{d} \cdot \frac{\left(M \cdot M\right) \cdot -0.125}{\ell}, h, 1\right) \cdot \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      -2e-131)
   (* (- (sqrt (/ 1.0 (* l h)))) d)
   (* (/ (fabs d) (sqrt (* l h))) 1.0)))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= -2e-131) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	}
	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) :: tmp
    if (((1.0d0 - ((h / l) * ((((d_1 * m) / (2.0d0 * d)) ** 2.0d0) * (1.0d0 / 2.0d0)))) * (((d / l) ** (1.0d0 / 2.0d0)) * ((d / h) ** (1.0d0 / 2.0d0)))) <= (-2d-131)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 - ((h / l) * (Math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)))) <= -2e-131) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if ((1.0 - ((h / l) * (math.pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))) <= -2e-131:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (Float64(Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= -2e-131)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (((1.0 - ((h / l) * ((((D * M) / (2.0 * d)) ^ 2.0) * (1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)))) <= -2e-131)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[N[(N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-131], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $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)))) < -2e-131

   1. Initial program 87.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 0.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 5.9

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 5.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   10. Step-by-step derivation

   11. Applied rewrites 8.6%

       \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -2e-131 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 56.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 56.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 56.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.5%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      5. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot 1 \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot 1 \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      20. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      21. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      22. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      23. lower-sqrt.f64 68.6

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   9. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-131}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)\\ \mathbf{if}\;d \leq -2.36 \cdot 10^{-113}:\\ \;\;\;\;\left(\left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}\right) \cdot {\left(-h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{\ell}} \cdot t\_0}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0)))
   (if (<= d -2.36e-113)
     (* (* (* t_0 (sqrt (/ d l))) (sqrt (- d))) (pow (- h) -0.5))
     (if (<= d 4.2e-138)
       (*
        (/ (fabs d) (sqrt (* l h)))
        (fma (* (* 0.25 (* (/ M d) D)) (* (/ 0.5 d) M)) (* (/ (- h) l) D) 1.0))
       (/ (* (/ (fabs d) (sqrt l)) t_0) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0);
	double tmp;
	if (d <= -2.36e-113) {
		tmp = ((t_0 * sqrt((d / l))) * sqrt(-d)) * pow(-h, -0.5);
	} else if (d <= 4.2e-138) {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((0.25 * ((M / d) * D)) * ((0.5 / d) * M)), ((-h / l) * D), 1.0);
	} else {
		tmp = ((fabs(d) / sqrt(l)) * t_0) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0)
	tmp = 0.0
	if (d <= -2.36e-113)
		tmp = Float64(Float64(Float64(t_0 * sqrt(Float64(d / l))) * sqrt(Float64(-d))) * (Float64(-h) ^ -0.5));
	elseif (d <= 4.2e-138)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(0.25 * Float64(Float64(M / d) * D)) * Float64(Float64(0.5 / d) * M)), Float64(Float64(Float64(-h) / l) * D), 1.0));
	else
		tmp = Float64(Float64(Float64(abs(d) / sqrt(l)) * t_0) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]}, If[LessEqual[d, -2.36e-113], N[(N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] * N[Power[(-h), -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-138], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.25 * N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[(N[((-h) / l), $MachinePrecision] * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.36e-113

   1. Initial program 78.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 78.7

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 78.7%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 89.1%

      \[\leadsto \color{blue}{{\left(-h\right)}^{-0.5} \cdot \left(\sqrt{-d} \cdot \left(\mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\right)} \]

   if -2.36e-113 < d < 4.19999999999999972e-138

   1. Initial program 41.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}} + 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}} + 1\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\frac{\ell}{h}}} + 1\right) \]

      9. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot h} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell}, h, 1\right)} \]

   4. Applied rewrites 42.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)} \]

   6. Applied rewrites 36.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right), \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right) + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \color{blue}{\left(\left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   8. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{M}{d} \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 4.19999999999999972e-138 < d

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 74.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 74.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right)}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.36 \cdot 10^{-113}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{-d}\right) \cdot {\left(-h\right)}^{-0.5}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{-\ell}}}{\sqrt{-h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -6.5e-157)
   (*
    (/ (/ (fabs d) (sqrt (- l))) (sqrt (- h)))
    (- 1.0 (* (/ h l) (* (pow (/ (* D M) (* 2.0 d)) 2.0) (/ 1.0 2.0)))))
   (if (<= d 4.2e-138)
     (*
      (/ (fabs d) (sqrt (* l h)))
      (fma (* (* 0.25 (* (/ M d) D)) (* (/ 0.5 d) M)) (* (/ (- h) l) D) 1.0))
     (/
      (*
       (/ (fabs d) (sqrt l))
       (fma (/ (* (pow (* (* (/ 0.5 d) D) M) 2.0) -0.5) l) h 1.0))
      (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -6.5e-157) {
		tmp = ((fabs(d) / sqrt(-l)) / sqrt(-h)) * (1.0 - ((h / l) * (pow(((D * M) / (2.0 * d)), 2.0) * (1.0 / 2.0))));
	} else if (d <= 4.2e-138) {
		tmp = (fabs(d) / sqrt((l * h))) * fma(((0.25 * ((M / d) * D)) * ((0.5 / d) * M)), ((-h / l) * D), 1.0);
	} else {
		tmp = ((fabs(d) / sqrt(l)) * fma(((pow((((0.5 / d) * D) * M), 2.0) * -0.5) / l), h, 1.0)) / sqrt(h);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -6.5e-157)
		tmp = Float64(Float64(Float64(abs(d) / sqrt(Float64(-l))) / sqrt(Float64(-h))) * Float64(1.0 - Float64(Float64(h / l) * Float64((Float64(Float64(D * M) / Float64(2.0 * d)) ^ 2.0) * Float64(1.0 / 2.0)))));
	elseif (d <= 4.2e-138)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * fma(Float64(Float64(0.25 * Float64(Float64(M / d) * D)) * Float64(Float64(0.5 / d) * M)), Float64(Float64(Float64(-h) / l) * D), 1.0));
	else
		tmp = Float64(Float64(Float64(abs(d) / sqrt(l)) * fma(Float64(Float64((Float64(Float64(Float64(0.5 / d) * D) * M) ^ 2.0) * -0.5) / l), h, 1.0)) / sqrt(h));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -6.5e-157], N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D * M), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e-138], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.25 * N[(N[(M / d), $MachinePrecision] * D), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / d), $MachinePrecision] * M), $MachinePrecision]), $MachinePrecision] * N[(N[((-h) / l), $MachinePrecision] * D), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(N[(N[(0.5 / d), $MachinePrecision] * D), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / l), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.5000000000000002e-157

   1. Initial program 80.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 80.2

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 80.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. frac-2neg N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{neg}\left(d\right)}{\color{blue}{-h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{d}{\ell}}{-h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{d}{\ell}}}{\sqrt{-h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \frac{d}{\ell}}}{\sqrt{-h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 73.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(-d\right) \cdot \frac{d}{\ell}}}{\sqrt{-h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-d\right) \cdot \frac{d}{\ell}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-d\right) \cdot \frac{d}{\ell}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-d\right) \cdot \color{blue}{\frac{d}{\ell}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. frac-2neg N/A

         \[\leadsto \frac{\sqrt{\left(-d\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(\ell\right)}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-d\right) \cdot \frac{\color{blue}{-d}}{\mathsf{neg}\left(\ell\right)}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{\left(-d\right) \cdot \left(-d\right)}{\mathsf{neg}\left(\ell\right)}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{\left(-d\right) \cdot \left(-d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \cdot \left(-d\right)}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{\left(\mathsf{neg}\left(d\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. sqr-neg N/A

          \[\leadsto \frac{\frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. rem-sqrt-square N/A

          \[\leadsto \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lift-fabs.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left|d\right|}}{\sqrt{\mathsf{neg}\left(\ell\right)}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{\mathsf{neg}\left(\ell\right)}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{\left|d\right|}{\color{blue}{\sqrt{\mathsf{neg}\left(\ell\right)}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. lower-neg.f64 87.6

          \[\leadsto \frac{\frac{\left|d\right|}{\sqrt{\color{blue}{-\ell}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 87.6%

      \[\leadsto \frac{\color{blue}{\frac{\left|d\right|}{\sqrt{-\ell}}}}{\sqrt{-h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   if -6.5000000000000002e-157 < d < 4.19999999999999972e-138

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \]

      2. sub-neg N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right) + 1\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)\right) + 1\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \frac{h}{\ell}} + 1\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{h}{\ell}} + 1\right) \]

      7. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{h}}} + 1\right) \]

      8. un-div-inv N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\frac{\ell}{h}}} + 1\right) \]

      9. associate-/r/ N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell} \cdot h} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}{\ell}, h, 1\right)} \]

   4. Applied rewrites 35.6%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-0.5 \cdot {\left(\frac{\frac{d}{D}}{M} \cdot 2\right)}^{-2}}{\ell}, h, 1\right)} \]

   6. Applied rewrites 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right), \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{0.5}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}, \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right) + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \color{blue}{\left(\left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\right)} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} + \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{M}{d} \cdot \left(\frac{1}{4} \cdot D\right)\right) \cdot \left(\left(\frac{-h}{\ell} \cdot D\right) \cdot \left(\frac{\frac{1}{2}}{d} \cdot M\right)\right) + 1\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}} \]

   8. Applied rewrites 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\frac{M}{d} \cdot D\right) \cdot 0.25\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right) \cdot \frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \]

   if 4.19999999999999972e-138 < d

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 74.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 74.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. unpow1/2 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{d}{h}}}}{\sqrt{\frac{\ell}{d}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{h}}}{\color{blue}{\sqrt{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\frac{d}{h}}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. un-div-inv N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h} \cdot \frac{1}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \frac{1}{\color{blue}{\frac{\ell}{d}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. clear-num N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. associate-*l/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot \frac{d}{\ell}}{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. pow1/2 N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{{h}^{\frac{1}{2}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{{h}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. pow-flip N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\color{blue}{\frac{1}{{h}^{\frac{-1}{2}}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. metadata-eval N/A

          \[\leadsto \frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\color{blue}{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      21. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\frac{1}{{h}^{\left(\frac{-1}{2}\right)}}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot \frac{d}{\ell}}}{\sqrt{h}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 90.5%

      \[\leadsto \color{blue}{\frac{\frac{\left|d\right|}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{-0.5 \cdot {\left(M \cdot \left(D \cdot \frac{0.5}{d}\right)\right)}^{2}}{\ell}, h, 1\right)}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{-\ell}}}{\sqrt{-h}} \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{D \cdot M}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(0.25 \cdot \left(\frac{M}{d} \cdot D\right)\right) \cdot \left(\frac{0.5}{d} \cdot M\right), \frac{-h}{\ell} \cdot D, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left|d\right|}{\sqrt{\ell}} \cdot \mathsf{fma}\left(\frac{{\left(\left(\frac{0.5}{d} \cdot D\right) \cdot M\right)}^{2} \cdot -0.5}{\ell}, h, 1\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 5e-137)
   (* (/ (fabs d) (sqrt (* l h))) 1.0)
   (/ d (* (sqrt h) (sqrt l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 5e-137) {
		tmp = (fabs(d) / sqrt((l * h))) * 1.0;
	} else {
		tmp = d / (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 (d <= 5d-137) then
        tmp = (abs(d) / sqrt((l * h))) * 1.0d0
    else
        tmp = d / (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 (d <= 5e-137) {
		tmp = (Math.abs(d) / Math.sqrt((l * h))) * 1.0;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 5e-137:
		tmp = (math.fabs(d) / math.sqrt((l * h))) * 1.0
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 5e-137)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(l * h))) * 1.0);
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 5e-137)
		tmp = (abs(d) / sqrt((l * h))) * 1.0;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 5e-137], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.00000000000000001e-137

   1. Initial program 62.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\frac{1}{2}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. unpow1/2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. clear-num N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{\frac{\ell}{d}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-/.f64 62.9

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\color{blue}{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Applied rewrites 62.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   5. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   6. Step-by-step derivation

   7. Applied rewrites 34.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot \color{blue}{1} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right)} \cdot 1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      3. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      5. unpow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{1}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{1}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      7. metadata-eval N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{\frac{\ell}{d}}}\right) \cdot 1 \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{d}}}}\right) \cdot 1 \]

      9. sqrt-div N/A

         \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{1}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{1}{\color{blue}{\frac{\ell}{d}}}}\right) \cdot 1 \]

      11. clear-num N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      12. lift-/.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\frac{d}{\ell}}}\right) \cdot 1 \]

      13. sqrt-unprod N/A

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \cdot 1 \]

      14. lift-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d}{h}} \cdot \frac{d}{\ell}} \cdot 1 \]

      15. lift-/.f64 N/A

          \[\leadsto \sqrt{\frac{d}{h} \cdot \color{blue}{\frac{d}{\ell}}} \cdot 1 \]

      16. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot 1 \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d \cdot d}{\color{blue}{\ell \cdot h}}} \cdot 1 \]

      19. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      20. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{d \cdot d}}{\sqrt{\ell \cdot h}}} \cdot 1 \]

      21. rem-sqrt-square N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      22. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{\ell \cdot h}} \cdot 1 \]

      23. lower-sqrt.f64 40.4

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   9. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \cdot 1 \]

   if 5.00000000000000001e-137 < d

   1. Initial program 75.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 52.3

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 63.1%

       \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 5 \cdot 10^{-137}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{-238}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -6e-238) (* (- (sqrt (/ 1.0 (* l h)))) d) (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e-238) {
		tmp = -sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / sqrt((l * h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-6d-238)) then
        tmp = -sqrt((1.0d0 / (l * h))) * d
    else
        tmp = d / sqrt((l * h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -6e-238) {
		tmp = -Math.sqrt((1.0 / (l * h))) * d;
	} else {
		tmp = d / Math.sqrt((l * h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -6e-238:
		tmp = -math.sqrt((1.0 / (l * h))) * d
	else:
		tmp = d / math.sqrt((l * h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -6e-238)
		tmp = Float64(Float64(-sqrt(Float64(1.0 / Float64(l * h)))) * d);
	else
		tmp = Float64(d / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -6e-238)
		tmp = -sqrt((1.0 / (l * h))) * d;
	else
		tmp = d / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -6e-238], N[((-N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * d), $MachinePrecision], N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.9999999999999999e-238

   1. Initial program 68.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.0%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 5.3

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 5.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   9. Taylor expanded in h around -inf

      \[\leadsto \left(\sqrt{\frac{1}{h \cdot \ell}} \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot d \]
   10. Step-by-step derivation

   11. Applied rewrites 40.6%

       \[\leadsto \left(-\sqrt{\frac{1}{\ell \cdot h}}\right) \cdot d \]

   if -5.9999999999999999e-238 < l

   1. Initial program 66.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.2%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

      8. remove-double-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

      13. lower-*.f64 48.7

          \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   8. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.7%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 26.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 40.1%

   \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{1} \]

6. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot \left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot d}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right)\right) \]

   6. rem-square-sqrt N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{-1} \cdot d\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(d\right)\right)}\right)\right) \]

   8. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{d} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot d} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot d \]

   11. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot d \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

   13. lower-*.f64 28.3

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot d \]

8. Applied rewrites 28.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
9. Step-by-step derivation

10. Applied rewrites 28.3%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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)))))