Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.7% → 75.3%

Time: 21.1s

Alternatives: 20

Speedup: 2.9×

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: 66.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.04 \cdot 10^{+158}:\\ \;\;\;\;\frac{t\_1 \cdot \left(t\_2 \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-\ell}} \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 8.5 \cdot 10^{+116}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_2\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (fma (/ (* M D) (* d 4.0)) (* (/ (* M D) d) (/ (* h -0.5) l)) 1.0))
        (t_1 (sqrt (- d)))
        (t_2 (sqrt (/ d l))))
   (if (<= h -1.04e+158)
     (/
      (*
       t_1
       (*
        t_2
        (fma (/ (* M (* D (* M D))) (* 4.0 (* d d))) (* (/ h l) -0.5) 1.0)))
      (sqrt (- h)))
     (if (<= h -1e-310)
       (* (* (/ t_1 (sqrt (- l))) t_0) (sqrt (/ d h)))
       (if (<= h 8.5e+116)
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d h) (/ 1.0 2.0)) (/ (sqrt d) (sqrt l))))
         (* (* t_0 t_2) (/ (sqrt d) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (d * 4.0)), (((M * D) / d) * ((h * -0.5) / l)), 1.0);
	double t_1 = sqrt(-d);
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -1.04e+158) {
		tmp = (t_1 * (t_2 * fma(((M * (D * (M * D))) / (4.0 * (d * d))), ((h / l) * -0.5), 1.0))) / sqrt(-h);
	} else if (h <= -1e-310) {
		tmp = ((t_1 / sqrt(-l)) * t_0) * sqrt((d / h));
	} else if (h <= 8.5e+116) {
		tmp = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / h), (1.0 / 2.0)) * (sqrt(d) / sqrt(l)));
	} else {
		tmp = (t_0 * t_2) * (sqrt(d) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(d * 4.0)), Float64(Float64(Float64(M * D) / d) * Float64(Float64(h * -0.5) / l)), 1.0)
	t_1 = sqrt(Float64(-d))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1.04e+158)
		tmp = Float64(Float64(t_1 * Float64(t_2 * fma(Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(4.0 * Float64(d * d))), Float64(Float64(h / l) * -0.5), 1.0))) / sqrt(Float64(-h)));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * t_0) * sqrt(Float64(d / h)));
	elseif (h <= 8.5e+116)
		tmp = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(d) / sqrt(l))));
	else
		tmp = Float64(Float64(t_0 * t_2) * Float64(sqrt(d) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1.04e+158], N[(N[(t$95$1 * N[(t$95$2 * N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 8.5e+116], N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$2), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -1.04000000000000004e158

   1. Initial program 51.6%

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

   4. Applied rewrites 64.6%

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

   if -1.04000000000000004e158 < h < -9.999999999999969e-311

   1. Initial program 67.1%

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

   4. Applied rewrites 48.5%

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

   6. Applied rewrites 48.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 82.4

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

   10. Applied rewrites 82.4%

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

   if -9.999999999999969e-311 < h < 8.5000000000000002e116

   1. Initial program 73.9%

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

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 83.0

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

   4. Applied rewrites 83.0%

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

   if 8.5000000000000002e116 < h

   1. Initial program 62.1%

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

   4. Applied rewrites 47.2%

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

   6. Applied rewrites 44.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 74.0

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

   10. Applied rewrites 74.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.04 \cdot 10^{+158}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;h \leq 8.5 \cdot 10^{+116}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t\_0 \cdot \left(t\_2 \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot 4\right)}, \frac{M \cdot D}{d}, 1\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{d}}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+234}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (* h l)))
        (t_2 (sqrt (/ d l)))
        (t_3
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_3 -2e-167)
     (*
      t_0
      (*
       t_2
       (fma (/ (* (* M D) (* h -0.5)) (* l (* d 4.0))) (/ (* M D) d) 1.0)))
     (if (<= t_3 1e-291)
       (/ 1.0 (/ t_1 d))
       (if (<= t_3 4e+234) (* t_0 t_2) (/ (sqrt (* d d)) t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((h * l));
	double t_2 = sqrt((d / l));
	double t_3 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_3 <= -2e-167) {
		tmp = t_0 * (t_2 * fma((((M * D) * (h * -0.5)) / (l * (d * 4.0))), ((M * D) / d), 1.0));
	} else if (t_3 <= 1e-291) {
		tmp = 1.0 / (t_1 / d);
	} else if (t_3 <= 4e+234) {
		tmp = t_0 * t_2;
	} else {
		tmp = sqrt((d * d)) / t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(h * l))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 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_3 <= -2e-167)
		tmp = Float64(t_0 * Float64(t_2 * fma(Float64(Float64(Float64(M * D) * Float64(h * -0.5)) / Float64(l * Float64(d * 4.0))), Float64(Float64(M * D) / d), 1.0)));
	elseif (t_3 <= 1e-291)
		tmp = Float64(1.0 / Float64(t_1 / d));
	elseif (t_3 <= 4e+234)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(sqrt(Float64(d * d)) / t_1);
	end
	return 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[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $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$3, -2e-167], N[(t$95$0 * N[(t$95$2 * N[(N[(N[(N[(M * D), $MachinePrecision] * N[(h * -0.5), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-291], N[(1.0 / N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+234], N[(t$95$0 * t$95$2), $MachinePrecision], N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-167

   1. Initial program 91.1%

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

   4. Applied rewrites 60.9%

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

   6. Applied rewrites 57.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 83.5%

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

   if -2e-167 < (*.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)))) < 9.99999999999999962e-292

   1. Initial program 41.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.9%

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

   if 9.99999999999999962e-292 < (*.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)))) < 4.00000000000000007e234

   1. Initial program 98.6%

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if 4.00000000000000007e234 < (*.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 19.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   9. Applied rewrites 27.6%

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

   11. Applied rewrites 32.2%

       \[\leadsto \frac{\sqrt{\left(d \cdot d\right) \cdot 1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot \left(h \cdot -0.5\right)}{\ell \cdot \left(d \cdot 4\right)}, \frac{M \cdot D}{d}, 1\right)\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{h \cdot \ell}\\ t_3 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-167}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(\frac{M}{d}, \left(D \cdot 0.25\right) \cdot \frac{M \cdot \left(D \cdot \left(h \cdot -0.5\right)\right)}{d \cdot \ell}, 1\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{d}}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+234}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (* h l)))
        (t_3
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_3 -2e-167)
     (*
      t_0
      (*
       t_1
       (fma (/ M d) (* (* D 0.25) (/ (* M (* D (* h -0.5))) (* d l))) 1.0)))
     (if (<= t_3 1e-291)
       (/ 1.0 (/ t_2 d))
       (if (<= t_3 4e+234) (* t_0 t_1) (/ (sqrt (* d d)) t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((h * l));
	double t_3 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_3 <= -2e-167) {
		tmp = t_0 * (t_1 * fma((M / d), ((D * 0.25) * ((M * (D * (h * -0.5))) / (d * l))), 1.0));
	} else if (t_3 <= 1e-291) {
		tmp = 1.0 / (t_2 / d);
	} else if (t_3 <= 4e+234) {
		tmp = t_0 * t_1;
	} else {
		tmp = sqrt((d * d)) / t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(h * l))
	t_3 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 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_3 <= -2e-167)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(M / d), Float64(Float64(D * 0.25) * Float64(Float64(M * Float64(D * Float64(h * -0.5))) / Float64(d * l))), 1.0)));
	elseif (t_3 <= 1e-291)
		tmp = Float64(1.0 / Float64(t_2 / d));
	elseif (t_3 <= 4e+234)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(sqrt(Float64(d * d)) / t_2);
	end
	return 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[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $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$3, -2e-167], N[(t$95$0 * N[(t$95$1 * N[(N[(M / d), $MachinePrecision] * N[(N[(D * 0.25), $MachinePrecision] * N[(N[(M * N[(D * N[(h * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-291], N[(1.0 / N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+234], N[(t$95$0 * t$95$1), $MachinePrecision], N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e-167

   1. Initial program 91.1%

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

   4. Applied rewrites 60.9%

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

   6. Applied rewrites 57.3%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 92.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 74.3%

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

   if -2e-167 < (*.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)))) < 9.99999999999999962e-292

   1. Initial program 41.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 77.8%

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

   9. Applied rewrites 77.9%

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

   if 9.99999999999999962e-292 < (*.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)))) < 4.00000000000000007e234

   1. Initial program 98.6%

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if 4.00000000000000007e234 < (*.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 19.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   9. Applied rewrites 27.6%

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

   11. Applied rewrites 32.2%

       \[\leadsto \frac{\sqrt{\left(d \cdot d\right) \cdot 1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M}{d}, \left(D \cdot 0.25\right) \cdot \frac{M \cdot \left(D \cdot \left(h \cdot -0.5\right)\right)}{d \cdot \ell}, 1\right)\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-151}:\\ \;\;\;\;t\_0 \cdot \left(t\_2 \cdot \mathsf{fma}\left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}, \frac{h}{d} \cdot -0.125, 1\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{d}}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+234}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h)))
        (t_1 (sqrt (* h l)))
        (t_2 (sqrt (/ d l)))
        (t_3
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_3 -2e-151)
     (*
      t_0
      (* t_2 (fma (/ (* M (* M (* D D))) (* d l)) (* (/ h d) -0.125) 1.0)))
     (if (<= t_3 1e-291)
       (/ 1.0 (/ t_1 d))
       (if (<= t_3 4e+234) (* t_0 t_2) (/ (sqrt (* d d)) t_1))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = sqrt((h * l));
	double t_2 = sqrt((d / l));
	double t_3 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_3 <= -2e-151) {
		tmp = t_0 * (t_2 * fma(((M * (M * (D * D))) / (d * l)), ((h / d) * -0.125), 1.0));
	} else if (t_3 <= 1e-291) {
		tmp = 1.0 / (t_1 / d);
	} else if (t_3 <= 4e+234) {
		tmp = t_0 * t_2;
	} else {
		tmp = sqrt((d * d)) / t_1;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = sqrt(Float64(h * l))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 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_3 <= -2e-151)
		tmp = Float64(t_0 * Float64(t_2 * fma(Float64(Float64(M * Float64(M * Float64(D * D))) / Float64(d * l)), Float64(Float64(h / d) * -0.125), 1.0)));
	elseif (t_3 <= 1e-291)
		tmp = Float64(1.0 / Float64(t_1 / d));
	elseif (t_3 <= 4e+234)
		tmp = Float64(t_0 * t_2);
	else
		tmp = Float64(sqrt(Float64(d * d)) / t_1);
	end
	return 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[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $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$3, -2e-151], N[(t$95$0 * N[(t$95$2 * N[(N[(N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $MachinePrecision] * N[(N[(h / d), $MachinePrecision] * -0.125), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-291], N[(1.0 / N[(t$95$1 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+234], N[(t$95$0 * t$95$2), $MachinePrecision], N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-151

   1. Initial program 91.0%

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

   4. Applied rewrites 61.6%

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

   6. Applied rewrites 57.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 67.3

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

   10. Applied rewrites 67.3%

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

   if -1.9999999999999999e-151 < (*.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)))) < 9.99999999999999962e-292

   1. Initial program 45.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 72.5%

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

   9. Applied rewrites 72.6%

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

   if 9.99999999999999962e-292 < (*.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)))) < 4.00000000000000007e234

   1. Initial program 98.6%

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if 4.00000000000000007e234 < (*.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 19.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   9. Applied rewrites 27.6%

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

   11. Applied rewrites 32.2%

       \[\leadsto \frac{\sqrt{\left(d \cdot d\right) \cdot 1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{d \cdot \ell}, \frac{h}{d} \cdot -0.125, 1\right)\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{h \cdot \ell}\\ t_3 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-151}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \mathsf{fma}\left(h \cdot -0.5, \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\left(d \cdot 4\right) \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{elif}\;t\_3 \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{d}}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+234}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (* h l)))
        (t_3
         (*
          (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_3 -2e-151)
     (*
      t_0
      (*
       t_1
       (fma (* h -0.5) (/ (* M (* M (* D D))) (* (* d 4.0) (* d l))) 1.0)))
     (if (<= t_3 1e-291)
       (/ 1.0 (/ t_2 d))
       (if (<= t_3 4e+234) (* t_1 t_0) (/ (sqrt (* d d)) t_2))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((h * l));
	double t_3 = (1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_3 <= -2e-151) {
		tmp = t_0 * (t_1 * fma((h * -0.5), ((M * (M * (D * D))) / ((d * 4.0) * (d * l))), 1.0));
	} else if (t_3 <= 1e-291) {
		tmp = 1.0 / (t_2 / d);
	} else if (t_3 <= 4e+234) {
		tmp = t_1 * t_0;
	} else {
		tmp = sqrt((d * d)) / t_2;
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(h * l))
	t_3 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 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_3 <= -2e-151)
		tmp = Float64(t_0 * Float64(t_1 * fma(Float64(h * -0.5), Float64(Float64(M * Float64(M * Float64(D * D))) / Float64(Float64(d * 4.0) * Float64(d * l))), 1.0)));
	elseif (t_3 <= 1e-291)
		tmp = Float64(1.0 / Float64(t_2 / d));
	elseif (t_3 <= 4e+234)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(sqrt(Float64(d * d)) / t_2);
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $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$3, -2e-151], N[(t$95$0 * N[(t$95$1 * N[(N[(h * -0.5), $MachinePrecision] * N[(N[(M * N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-291], N[(1.0 / N[(t$95$2 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+234], N[(t$95$1 * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.9999999999999999e-151

   1. Initial program 91.0%

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

   4. Applied rewrites 61.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 91.1

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

   6. Applied rewrites 91.1%

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

   8. Applied rewrites 61.4%

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

   if -1.9999999999999999e-151 < (*.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)))) < 9.99999999999999962e-292

   1. Initial program 45.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   7. Applied rewrites 72.5%

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

   9. Applied rewrites 72.6%

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

   if 9.99999999999999962e-292 < (*.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)))) < 4.00000000000000007e234

   1. Initial program 98.6%

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

   4. Applied rewrites 83.7%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 98.3

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

   9. Applied rewrites 98.3%

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

   if 4.00000000000000007e234 < (*.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 19.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   9. Applied rewrites 27.6%

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

   11. Applied rewrites 32.2%

       \[\leadsto \frac{\sqrt{\left(d \cdot d\right) \cdot 1}}{\color{blue}{\sqrt{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -2 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \mathsf{fma}\left(h \cdot -0.5, \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\left(d \cdot 4\right) \cdot \left(d \cdot \ell\right)}, 1\right)\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{h \cdot \ell}}{d}}\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))))
   (if (<=
        (*
         (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
         (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
        INFINITY)
     (*
      (sqrt (/ d h))
      (*
       (fma (/ (* M D) (* d 4.0)) (* (/ (* M D) d) (/ (* h -0.5) l)) 1.0)
       t_0))
     (*
      (*
       t_0
       (+ 1.0 (/ (* (* h -0.5) (* M (* D (* M D)))) (* l (* d (* d 4.0))))))
      (/ (sqrt (- d)) (sqrt (- h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= ((double) INFINITY)) {
		tmp = sqrt((d / h)) * (fma(((M * D) / (d * 4.0)), (((M * D) / d) * ((h * -0.5) / l)), 1.0) * t_0);
	} else {
		tmp = (t_0 * (1.0 + (((h * -0.5) * (M * (D * (M * D)))) / (l * (d * (d * 4.0)))))) * (sqrt(-d) / sqrt(-h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= Inf)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(fma(Float64(Float64(M * D) / Float64(d * 4.0)), Float64(Float64(Float64(M * D) / d) * Float64(Float64(h * -0.5) / l)), 1.0) * t_0));
	else
		tmp = Float64(Float64(t_0 * Float64(1.0 + Float64(Float64(Float64(h * -0.5) * Float64(M * Float64(D * Float64(M * D)))) / Float64(l * Float64(d * Float64(d * 4.0)))))) * Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $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], Infinity], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(1.0 + N[(N[(N[(h * -0.5), $MachinePrecision] * N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(d * N[(d * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.1%

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

   4. Applied rewrites 61.3%

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

   6. Applied rewrites 57.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 83.5%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 11.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-neg.f64 22.4

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

   8. Applied rewrites 22.4%

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

4. Final simplification 71.4%

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

Alternative 7: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<=
      (*
       (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
       (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))
      4e+234)
   (*
    (sqrt (/ d h))
    (*
     (fma (/ (* M D) (* d 4.0)) (* (/ (* M D) d) (/ (* h -0.5) l)) 1.0)
     (sqrt (/ d l))))
   (/ (sqrt (* d d)) (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (((1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)))) <= 4e+234) {
		tmp = sqrt((d / h)) * (fma(((M * D) / (d * 4.0)), (((M * D) / d) * ((h * -0.5) / l)), 1.0) * sqrt((d / l)));
	} else {
		tmp = sqrt((d * d)) / sqrt((h * l));
	}
	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(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0)))) <= 4e+234)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(fma(Float64(Float64(M * D) / Float64(d * 4.0)), Float64(Float64(Float64(M * D) / d) * Float64(Float64(h * -0.5) / l)), 1.0) * sqrt(Float64(d / l))));
	else
		tmp = Float64(sqrt(Float64(d * d)) / sqrt(Float64(h * l)));
	end
	return 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[(M * D), $MachinePrecision] / N[(d * 2.0), $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], 4e+234], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(d * d), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.00000000000000007e234

   1. Initial program 90.4%

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

   4. Applied rewrites 65.7%

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

   6. Applied rewrites 62.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 91.0%

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

   if 4.00000000000000007e234 < (*.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 19.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   7. Applied rewrites 25.6%

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

   9. Applied rewrites 27.6%

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

   11. Applied rewrites 32.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq 4 \cdot 10^{+234}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;d \leq -1.95 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(t\_1 \cdot \frac{1}{\sqrt{-h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot t\_0\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (+ 1.0 (* (/ h l) (* (pow (/ (* M D) (* d 2.0)) 2.0) (/ -1.0 2.0)))))
        (t_1 (sqrt (- d))))
   (if (<= d -1.95e-150)
     (* (* (* t_1 (/ 1.0 (sqrt (- h)))) (pow (/ d l) (/ 1.0 2.0))) t_0)
     (if (<= d -2e-310)
       (*
        (*
         (/ t_1 (sqrt (- l)))
         (fma (/ (* M D) (* d 4.0)) (* (/ (* M D) d) (/ (* h -0.5) l)) 1.0))
        (sqrt (/ d h)))
       (* t_0 (* (pow (/ d h) (/ 1.0 2.0)) (/ (sqrt d) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 + ((h / l) * (pow(((M * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)));
	double t_1 = sqrt(-d);
	double tmp;
	if (d <= -1.95e-150) {
		tmp = ((t_1 * (1.0 / sqrt(-h))) * pow((d / l), (1.0 / 2.0))) * t_0;
	} else if (d <= -2e-310) {
		tmp = ((t_1 / sqrt(-l)) * fma(((M * D) / (d * 4.0)), (((M * D) / d) * ((h * -0.5) / l)), 1.0)) * sqrt((d / h));
	} else {
		tmp = t_0 * (pow((d / h), (1.0 / 2.0)) * (sqrt(d) / sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (d <= -1.95e-150)
		tmp = Float64(Float64(Float64(t_1 * Float64(1.0 / sqrt(Float64(-h)))) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * t_0);
	elseif (d <= -2e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * fma(Float64(Float64(M * D) / Float64(d * 4.0)), Float64(Float64(Float64(M * D) / d) * Float64(Float64(h * -0.5) / l)), 1.0)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(t_0 * Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * Float64(sqrt(d) / sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[d, -1.95e-150], N[(N[(N[(t$95$1 * N[(1.0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, -2e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.9500000000000001e-150

   1. Initial program 69.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 82.7

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

   4. Applied rewrites 82.7%

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

   if -1.9500000000000001e-150 < d < -1.999999999999994e-310

   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

   4. Applied rewrites 16.2%

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

   6. Applied rewrites 15.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 45.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 67.9

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

   10. Applied rewrites 67.9%

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

   if -1.999999999999994e-310 < d

   1. Initial program 70.7%

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

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-sqrt.f64 77.4

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

   4. Applied rewrites 77.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-150}:\\ \;\;\;\;\left(\left(\sqrt{-d} \cdot \frac{1}{\sqrt{-h}}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right)\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right)\\ t_1 := \sqrt{-d}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -1.04 \cdot 10^{+158}:\\ \;\;\;\;\frac{t\_1 \cdot \left(t\_2 \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{t\_1}{\sqrt{-\ell}} \cdot t\_0\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_2\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (fma (/ (* M D) (* d 4.0)) (* (/ (* M D) d) (/ (* h -0.5) l)) 1.0))
        (t_1 (sqrt (- d)))
        (t_2 (sqrt (/ d l))))
   (if (<= h -1.04e+158)
     (/
      (*
       t_1
       (*
        t_2
        (fma (/ (* M (* D (* M D))) (* 4.0 (* d d))) (* (/ h l) -0.5) 1.0)))
      (sqrt (- h)))
     (if (<= h -1e-310)
       (* (* (/ t_1 (sqrt (- l))) t_0) (sqrt (/ d h)))
       (* (* t_0 t_2) (/ (sqrt d) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = fma(((M * D) / (d * 4.0)), (((M * D) / d) * ((h * -0.5) / l)), 1.0);
	double t_1 = sqrt(-d);
	double t_2 = sqrt((d / l));
	double tmp;
	if (h <= -1.04e+158) {
		tmp = (t_1 * (t_2 * fma(((M * (D * (M * D))) / (4.0 * (d * d))), ((h / l) * -0.5), 1.0))) / sqrt(-h);
	} else if (h <= -1e-310) {
		tmp = ((t_1 / sqrt(-l)) * t_0) * sqrt((d / h));
	} else {
		tmp = (t_0 * t_2) * (sqrt(d) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = fma(Float64(Float64(M * D) / Float64(d * 4.0)), Float64(Float64(Float64(M * D) / d) * Float64(Float64(h * -0.5) / l)), 1.0)
	t_1 = sqrt(Float64(-d))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (h <= -1.04e+158)
		tmp = Float64(Float64(t_1 * Float64(t_2 * fma(Float64(Float64(M * Float64(D * Float64(M * D))) / Float64(4.0 * Float64(d * d))), Float64(Float64(h / l) * -0.5), 1.0))) / sqrt(Float64(-h)));
	elseif (h <= -1e-310)
		tmp = Float64(Float64(Float64(t_1 / sqrt(Float64(-l))) * t_0) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(t_0 * t_2) * Float64(sqrt(d) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[(M * D), $MachinePrecision] / N[(d * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1.04e+158], N[(N[(t$95$1 * N[(t$95$2 * N[(N[(N[(M * N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -1e-310], N[(N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * t$95$2), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.04000000000000004e158

   1. Initial program 51.6%

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

   4. Applied rewrites 64.6%

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

   if -1.04000000000000004e158 < h < -9.999999999999969e-311

   1. Initial program 67.1%

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

   4. Applied rewrites 48.5%

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

   6. Applied rewrites 48.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 82.4

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

   10. Applied rewrites 82.4%

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

   if -9.999999999999969e-311 < h

   1. Initial program 70.7%

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

   4. Applied rewrites 51.8%

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

   6. Applied rewrites 52.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. frac-times N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

   8. Applied rewrites 70.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 76.8

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

   10. Applied rewrites 76.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.04 \cdot 10^{+158}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot \left(D \cdot \left(M \cdot D\right)\right)}{4 \cdot \left(d \cdot d\right)}, \frac{h}{\ell} \cdot -0.5, 1\right)\right)}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot D}{d \cdot 4}, \frac{M \cdot D}{d} \cdot \frac{h \cdot -0.5}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\sqrt{d}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}\\ t_1 := M \cdot \left(D \cdot D\right)\\ \mathbf{if}\;\ell \leq -3.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;t\_0 \cdot \left(\left(t\_1 \cdot \left(-M\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot -0.5, \frac{M \cdot t\_1}{\left(d \cdot 4\right) \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{t\_0}{d}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ h (* l (* l l))))) (t_1 (* M (* D D))))
   (if (<= l -3.15e-40)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l 3.8e-305)
       (* t_0 (* (* t_1 (- M)) (/ -0.125 d)))
       (if (<= l 1.6e-26)
         (*
          (fma (* h -0.5) (/ (* M t_1) (* (* d 4.0) (* d l))) 1.0)
          (/ d (sqrt (* h l))))
         (if (<= l 1.2e+115)
           (fma
            -0.125
            (* (* D (* D (* M M))) (/ t_0 d))
            (* d (sqrt (/ 1.0 (* h l)))))
           (/ d (* (sqrt l) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((h / (l * (l * l))));
	double t_1 = M * (D * D);
	double tmp;
	if (l <= -3.15e-40) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= 3.8e-305) {
		tmp = t_0 * ((t_1 * -M) * (-0.125 / d));
	} else if (l <= 1.6e-26) {
		tmp = fma((h * -0.5), ((M * t_1) / ((d * 4.0) * (d * l))), 1.0) * (d / sqrt((h * l)));
	} else if (l <= 1.2e+115) {
		tmp = fma(-0.125, ((D * (D * (M * M))) * (t_0 / d)), (d * sqrt((1.0 / (h * l)))));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(h / Float64(l * Float64(l * l))))
	t_1 = Float64(M * Float64(D * D))
	tmp = 0.0
	if (l <= -3.15e-40)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= 3.8e-305)
		tmp = Float64(t_0 * Float64(Float64(t_1 * Float64(-M)) * Float64(-0.125 / d)));
	elseif (l <= 1.6e-26)
		tmp = Float64(fma(Float64(h * -0.5), Float64(Float64(M * t_1) / Float64(Float64(d * 4.0) * Float64(d * l))), 1.0) * Float64(d / sqrt(Float64(h * l))));
	elseif (l <= 1.2e+115)
		tmp = fma(-0.125, Float64(Float64(D * Float64(D * Float64(M * M))) * Float64(t_0 / d)), Float64(d * sqrt(Float64(1.0 / Float64(h * l)))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.15e-40], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e-305], N[(t$95$0 * N[(N[(t$95$1 * (-M)), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.6e-26], N[(N[(N[(h * -0.5), $MachinePrecision] * N[(N[(M * t$95$1), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.2e+115], N[(-0.125 * N[(N[(D * N[(D * N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / d), $MachinePrecision]), $MachinePrecision] + N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -3.1500000000000001e-40

   1. Initial program 61.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 6.4%

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

   9. Applied rewrites 39.5%

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

   11. Applied rewrites 60.1%

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

   if -3.1500000000000001e-40 < l < 3.8e-305

   1. Initial program 62.1%

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 53.1%

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

   if 3.8e-305 < l < 1.6000000000000001e-26

   1. Initial program 78.9%

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

   4. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 78.8

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

   6. Applied rewrites 78.8%

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

   8. Applied rewrites 64.5%

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

   if 1.6000000000000001e-26 < l < 1.2e115

   1. Initial program 72.2%

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

   4. Applied rewrites 49.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in M around 0

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

      1. lower-fma.f64 N/A

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

   9. Applied rewrites 82.0%

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

   if 1.2e115 < l

   1. Initial program 52.0%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 45.4

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

   5. Applied rewrites 45.4%

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

   7. Applied rewrites 45.4%

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

   9. Applied rewrites 55.9%

      \[\leadsto \frac{d}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\left(M \cdot \left(D \cdot D\right)\right) \cdot \left(-M\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot -0.5, \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\left(d \cdot 4\right) \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \left(D \cdot \left(D \cdot \left(M \cdot M\right)\right)\right) \cdot \frac{\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}}}{d}, d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \left(D \cdot D\right)\\ \mathbf{if}\;\ell \leq -3.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(t\_0 \cdot \left(-M\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot -0.5, \frac{M \cdot t\_0}{\left(d \cdot 4\right) \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (* D D))))
   (if (<= l -3.15e-40)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l 3.8e-305)
       (* (sqrt (/ h (* l (* l l)))) (* (* t_0 (- M)) (/ -0.125 d)))
       (if (<= l 9.8e-95)
         (*
          (fma (* h -0.5) (/ (* M t_0) (* (* d 4.0) (* d l))) 1.0)
          (/ d (sqrt (* h l))))
         (* d (/ (/ 1.0 (sqrt h)) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D * D);
	double tmp;
	if (l <= -3.15e-40) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= 3.8e-305) {
		tmp = sqrt((h / (l * (l * l)))) * ((t_0 * -M) * (-0.125 / d));
	} else if (l <= 9.8e-95) {
		tmp = fma((h * -0.5), ((M * t_0) / ((d * 4.0) * (d * l))), 1.0) * (d / sqrt((h * l)));
	} else {
		tmp = d * ((1.0 / sqrt(h)) / sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(D * D))
	tmp = 0.0
	if (l <= -3.15e-40)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= 3.8e-305)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(t_0 * Float64(-M)) * Float64(-0.125 / d)));
	elseif (l <= 9.8e-95)
		tmp = Float64(fma(Float64(h * -0.5), Float64(Float64(M * t_0) / Float64(Float64(d * 4.0) * Float64(d * l))), 1.0) * Float64(d / sqrt(Float64(h * l))));
	else
		tmp = Float64(d * Float64(Float64(1.0 / sqrt(h)) / sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.15e-40], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.8e-305], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$0 * (-M)), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.8e-95], N[(N[(N[(h * -0.5), $MachinePrecision] * N[(N[(M * t$95$0), $MachinePrecision] / N[(N[(d * 4.0), $MachinePrecision] * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[(1.0 / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.1500000000000001e-40

   1. Initial program 61.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 6.4%

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

   9. Applied rewrites 39.5%

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

   11. Applied rewrites 60.1%

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

   if -3.1500000000000001e-40 < l < 3.8e-305

   1. Initial program 62.1%

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 53.1%

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

   if 3.8e-305 < l < 9.8e-95

   1. Initial program 80.4%

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

   4. Applied rewrites 64.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 80.4

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

   6. Applied rewrites 80.4%

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

   8. Applied rewrites 68.0%

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

   if 9.8e-95 < l

   1. Initial program 65.5%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 57.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\left(M \cdot \left(D \cdot D\right)\right) \cdot \left(-M\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(h \cdot -0.5, \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{\left(d \cdot 4\right) \cdot \left(d \cdot \ell\right)}, 1\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{\frac{1}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\left(M \cdot \left(D \cdot D\right)\right) \cdot \left(-M\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -3.15e-40)
   (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
   (if (<= l -5e-310)
     (* (sqrt (/ h (* l (* l l)))) (* (* (* M (* D D)) (- M)) (/ -0.125 d)))
     (/ d (* (sqrt l) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.15e-40) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -5e-310) {
		tmp = sqrt((h / (l * (l * l)))) * (((M * (D * D)) * -M) * (-0.125 / d));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-3.15d-40)) then
        tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h)
    else if (l <= (-5d-310)) then
        tmp = sqrt((h / (l * (l * l)))) * (((m * (d_1 * d_1)) * -m) * ((-0.125d0) / d))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3.15e-40) {
		tmp = (Math.sqrt(-d) * Math.sqrt((d / l))) / Math.sqrt(-h);
	} else if (l <= -5e-310) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (((M * (D * D)) * -M) * (-0.125 / d));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -3.15e-40:
		tmp = (math.sqrt(-d) * math.sqrt((d / l))) / math.sqrt(-h)
	elif l <= -5e-310:
		tmp = math.sqrt((h / (l * (l * l)))) * (((M * (D * D)) * -M) * (-0.125 / d))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -3.15e-40)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -5e-310)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(Float64(M * Float64(D * D)) * Float64(-M)) * Float64(-0.125 / d)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -3.15e-40)
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	elseif (l <= -5e-310)
		tmp = sqrt((h / (l * (l * l)))) * (((M * (D * D)) * -M) * (-0.125 / d));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -3.15e-40], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(M * N[(D * D), $MachinePrecision]), $MachinePrecision] * (-M)), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.1500000000000001e-40

   1. Initial program 61.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 6.4

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 6.4%

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

   9. Applied rewrites 39.5%

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

   11. Applied rewrites 60.1%

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

   if -3.1500000000000001e-40 < l < -4.999999999999985e-310

   1. Initial program 63.3%

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

   7. Applied rewrites 54.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 70.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 52.0%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.15 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(\left(\left(M \cdot \left(D \cdot D\right)\right) \cdot \left(-M\right)\right) \cdot \frac{-0.125}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;D \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{M \cdot \left(M \cdot D\right)}{d \cdot 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5.2e-8)
   (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
   (if (<= l -5e-310)
     (* D (* (sqrt (/ h (* l (* l l)))) (/ (* M (* M D)) (* d 8.0))))
     (/ d (* (sqrt l) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.2e-8) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -5e-310) {
		tmp = D * (sqrt((h / (l * (l * l)))) * ((M * (M * D)) / (d * 8.0)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5.2d-8)) then
        tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h)
    else if (l <= (-5d-310)) then
        tmp = d_1 * (sqrt((h / (l * (l * l)))) * ((m * (m * d_1)) / (d * 8.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.2e-8) {
		tmp = (Math.sqrt(-d) * Math.sqrt((d / l))) / Math.sqrt(-h);
	} else if (l <= -5e-310) {
		tmp = D * (Math.sqrt((h / (l * (l * l)))) * ((M * (M * D)) / (d * 8.0)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -5.2e-8:
		tmp = (math.sqrt(-d) * math.sqrt((d / l))) / math.sqrt(-h)
	elif l <= -5e-310:
		tmp = D * (math.sqrt((h / (l * (l * l)))) * ((M * (M * D)) / (d * 8.0)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5.2e-8)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -5e-310)
		tmp = Float64(D * Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(Float64(M * Float64(M * D)) / Float64(d * 8.0))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5.2e-8)
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	elseif (l <= -5e-310)
		tmp = D * (sqrt((h / (l * (l * l)))) * ((M * (M * D)) / (d * 8.0)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5.2e-8], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(D * N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(M * N[(M * D), $MachinePrecision]), $MachinePrecision] / N[(d * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.2000000000000002e-8

   1. Initial program 61.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 6.8

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

   5. Applied rewrites 6.8%

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

   7. Applied rewrites 6.8%

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

   9. Applied rewrites 41.3%

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

   11. Applied rewrites 61.0%

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

   if -5.2000000000000002e-8 < l < -4.999999999999985e-310

   1. Initial program 63.8%

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

   3. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

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

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

      13. distribute-rgt-neg-in N/A

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

      14. distribute-neg-frac N/A

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

   5. Applied rewrites 44.3%

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

   7. Applied rewrites 55.5%

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

   if -4.999999999999985e-310 < l

   1. Initial program 70.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 52.0%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;D \cdot \left(\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \frac{M \cdot \left(M \cdot D\right)}{d \cdot 8}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{h \cdot \ell}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* h l))))
   (if (<= l -5e-239)
     (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
     (if (<= l -5e-310)
       (* d (sqrt (sqrt (* t_0 t_0))))
       (/ d (* (sqrt l) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (l <= -5e-239) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (l <= -5e-310) {
		tmp = d * sqrt(sqrt((t_0 * t_0)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (h * l)
    if (l <= (-5d-239)) then
        tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h)
    else if (l <= (-5d-310)) then
        tmp = d * sqrt(sqrt((t_0 * t_0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 / (h * l);
	double tmp;
	if (l <= -5e-239) {
		tmp = (Math.sqrt(-d) * Math.sqrt((d / l))) / Math.sqrt(-h);
	} else if (l <= -5e-310) {
		tmp = d * Math.sqrt(Math.sqrt((t_0 * t_0)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 / (h * l)
	tmp = 0
	if l <= -5e-239:
		tmp = (math.sqrt(-d) * math.sqrt((d / l))) / math.sqrt(-h)
	elif l <= -5e-310:
		tmp = d * math.sqrt(math.sqrt((t_0 * t_0)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 / Float64(h * l))
	tmp = 0.0
	if (l <= -5e-239)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (l <= -5e-310)
		tmp = Float64(d * sqrt(sqrt(Float64(t_0 * t_0))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 / (h * l);
	tmp = 0.0;
	if (l <= -5e-239)
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	elseif (l <= -5e-310)
		tmp = d * sqrt(sqrt((t_0 * t_0)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-239], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e-239

   1. Initial program 59.5%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 6.1%

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

   9. Applied rewrites 33.7%

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

   11. Applied rewrites 49.0%

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

   if -5e-239 < l < -4.999999999999985e-310

   1. Initial program 74.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 31.6

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

   5. Applied rewrites 31.6%

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

   7. Applied rewrites 55.5%

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

   if -4.999999999999985e-310 < l

   1. Initial program 70.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 52.0%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\sqrt{\frac{1}{h \cdot \ell} \cdot \frac{1}{h \cdot \ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.56 \cdot 10^{+193}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.56e+193)
   (/ (* (sqrt (- d)) (sqrt (/ d l))) (sqrt (- h)))
   (if (<= h 4.6e-293)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (/ d (* (sqrt l) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.56e+193) {
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	} else if (h <= 4.6e-293) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1.56d+193)) then
        tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h)
    else if (h <= 4.6d-293) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.56e+193) {
		tmp = (Math.sqrt(-d) * Math.sqrt((d / l))) / Math.sqrt(-h);
	} else if (h <= 4.6e-293) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1.56e+193:
		tmp = (math.sqrt(-d) * math.sqrt((d / l))) / math.sqrt(-h)
	elif h <= 4.6e-293:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.56e+193)
		tmp = Float64(Float64(sqrt(Float64(-d)) * sqrt(Float64(d / l))) / sqrt(Float64(-h)));
	elseif (h <= 4.6e-293)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1.56e+193)
		tmp = (sqrt(-d) * sqrt((d / l))) / sqrt(-h);
	elseif (h <= 4.6e-293)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.56e+193], N[(N[(N[Sqrt[(-d)], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 4.6e-293], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.5599999999999999e193

   1. Initial program 55.2%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 6.1

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

   5. Applied rewrites 6.1%

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

   7. Applied rewrites 6.1%

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

   9. Applied rewrites 20.8%

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

   11. Applied rewrites 33.1%

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

   if -1.5599999999999999e193 < h < 4.5999999999999999e-293

   1. Initial program 64.8%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 4.5999999999999999e-293 < h

   1. Initial program 70.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 47.6%

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

   9. Applied rewrites 52.9%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.56 \cdot 10^{+193}:\\ \;\;\;\;\frac{\sqrt{-d} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-h}}\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1.56 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1.56e+193)
   (* (sqrt (/ d h)) (sqrt (/ d l)))
   (if (<= h 4.6e-293)
     (* (- d) (sqrt (/ 1.0 (* h l))))
     (/ d (* (sqrt l) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.56e+193) {
		tmp = sqrt((d / h)) * sqrt((d / l));
	} else if (h <= 4.6e-293) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1.56d+193)) then
        tmp = sqrt((d / h)) * sqrt((d / l))
    else if (h <= 4.6d-293) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1.56e+193) {
		tmp = Math.sqrt((d / h)) * Math.sqrt((d / l));
	} else if (h <= 4.6e-293) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1.56e+193:
		tmp = math.sqrt((d / h)) * math.sqrt((d / l))
	elif h <= 4.6e-293:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1.56e+193)
		tmp = Float64(sqrt(Float64(d / h)) * sqrt(Float64(d / l)));
	elseif (h <= 4.6e-293)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1.56e+193)
		tmp = sqrt((d / h)) * sqrt((d / l));
	elseif (h <= 4.6e-293)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1.56e+193], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 4.6e-293], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.5599999999999999e193

   1. Initial program 55.2%

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

   4. Applied rewrites 45.5%

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

   6. Applied rewrites 39.2%

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

   7. Taylor expanded in d around inf

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 33.1

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

   9. Applied rewrites 33.1%

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

   if -1.5599999999999999e193 < h < 4.5999999999999999e-293

   1. Initial program 64.8%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 49.3

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

   5. Applied rewrites 49.3%

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

   if 4.5999999999999999e-293 < h

   1. Initial program 70.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 47.6%

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

   9. Applied rewrites 52.9%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1.56 \cdot 10^{+193}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;h \leq 4.6 \cdot 10^{-293}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 8.6 \cdot 10^{-262}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d 8.6e-262)
   (* (- d) (sqrt (/ 1.0 (* h l))))
   (/ d (* (sqrt l) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 8.6e-262) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 8.6d-262) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 8.6e-262) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= 8.6e-262:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 8.6e-262)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 8.6e-262)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, 8.6e-262], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 8.6000000000000002e-262

   1. Initial program 62.0%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 40.5

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

   5. Applied rewrites 40.5%

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

   if 8.6000000000000002e-262 < d

   1. Initial program 71.8%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   7. Applied rewrites 49.5%

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

   9. Applied rewrites 54.2%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 8.6 \cdot 10^{-262}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-240}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -7.8e-240) (* (- d) (sqrt (/ 1.0 (* h l)))) (/ d (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.8e-240) {
		tmp = -d * sqrt((1.0 / (h * l)));
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-7.8d-240)) then
        tmp = -d * sqrt((1.0d0 / (h * l)))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -7.8e-240) {
		tmp = -d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -7.8e-240:
		tmp = -d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -7.8e-240)
		tmp = Float64(Float64(-d) * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -7.8e-240)
		tmp = -d * sqrt((1.0 / (h * l)));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -7.8e-240], N[((-d) * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.80000000000000029e-240

   1. Initial program 59.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in l around -inf

      \[\leadsto \color{blue}{\left(d \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot d\right)} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      2. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      3. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \cdot \left(-1 \cdot d\right) \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      10. lower-neg.f64 46.2

          \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(-d\right)} \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right)} \]

   if -7.80000000000000029e-240 < l

   1. Initial program 71.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 43.8

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 43.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{-240}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.1e-201) (sqrt (* d (/ d (* h l)))) (/ d (sqrt (* h l)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.1e-201) {
		tmp = sqrt((d * (d / (h * l))));
	} else {
		tmp = d / sqrt((h * l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.1d-201)) then
        tmp = sqrt((d * (d / (h * l))))
    else
        tmp = d / sqrt((h * l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.1e-201) {
		tmp = Math.sqrt((d * (d / (h * l))));
	} else {
		tmp = d / Math.sqrt((h * l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.1e-201:
		tmp = math.sqrt((d * (d / (h * l))))
	else:
		tmp = d / math.sqrt((h * l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.1e-201)
		tmp = sqrt(Float64(d * Float64(d / Float64(h * l))));
	else
		tmp = Float64(d / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.1e-201)
		tmp = sqrt((d * (d / (h * l))));
	else
		tmp = d / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.1e-201], N[Sqrt[N[(d * N[(d / N[(h * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.1e-201

   1. Initial program 68.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 7.0

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 7.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.8%

      \[\leadsto \sqrt{\frac{d \cdot d}{h \cdot \ell}} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.4%

       \[\leadsto \sqrt{\frac{d}{h \cdot \ell} \cdot d} \]

   if -1.1e-201 < d

   1. Initial program 65.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

      3. lower-/.f64 N/A

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      4. lower-*.f64 43.4

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

   5. Applied rewrites 43.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{d \cdot \frac{d}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{h \cdot \ell}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* h l))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((h * l));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((h * l))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((h * l));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((h * l))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(h * l)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((h * l));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
2. Add Preprocessing

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

   3. lower-/.f64 N/A

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

   4. lower-*.f64 28.3

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{h \cdot \ell}}} \]

5. Applied rewrites 28.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

7. Applied rewrites 28.7%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))))