Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 78.4%

Time: 15.6s

Alternatives: 16

Speedup: 1.8×

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: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \sqrt{-d}\\ t_1 := M\_m \cdot \frac{D}{d}\\ t_2 := \mathsf{fma}\left(\frac{t\_1}{\ell}, \frac{t\_1 \cdot -0.125}{{h}^{-1}}, 1\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;h \leq -2.9 \cdot 10^{-238}:\\ \;\;\;\;t\_2 \cdot \frac{t\_0}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M\_m} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_0}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (sqrt (- d)))
        (t_1 (* M_m (/ D d)))
        (t_2
         (*
          (fma (/ t_1 l) (/ (* t_1 -0.125) (pow h -1.0)) 1.0)
          (sqrt (/ d h)))))
   (if (<= h -2.9e-238)
     (* t_2 (/ t_0 (sqrt (- l))))
     (if (<= h -4e-310)
       (/
        (*
         (*
          (fma (* -0.5 (/ h l)) (pow (* (/ d M_m) (/ 2.0 D)) -2.0) 1.0)
          (sqrt (/ d l)))
         t_0)
        (sqrt (- h)))
       (* t_2 (/ (sqrt d) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = sqrt(-d);
	double t_1 = M_m * (D / d);
	double t_2 = fma((t_1 / l), ((t_1 * -0.125) / pow(h, -1.0)), 1.0) * sqrt((d / h));
	double tmp;
	if (h <= -2.9e-238) {
		tmp = t_2 * (t_0 / sqrt(-l));
	} else if (h <= -4e-310) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M_m) * (2.0 / D)), -2.0), 1.0) * sqrt((d / l))) * t_0) / sqrt(-h);
	} else {
		tmp = t_2 * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = sqrt(Float64(-d))
	t_1 = Float64(M_m * Float64(D / d))
	t_2 = Float64(fma(Float64(t_1 / l), Float64(Float64(t_1 * -0.125) / (h ^ -1.0)), 1.0) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (h <= -2.9e-238)
		tmp = Float64(t_2 * Float64(t_0 / sqrt(Float64(-l))));
	elseif (h <= -4e-310)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M_m) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * t_0) / sqrt(Float64(-h)));
	else
		tmp = Float64(t_2 * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 / l), $MachinePrecision] * N[(N[(t$95$1 * -0.125), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -2.9e-238], N[(t$95$2 * N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.8999999999999998e-238

   1. Initial program 69.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 73.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 75.7%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 83.0

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

   10. Applied rewrites 83.0%

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

   if -2.8999999999999998e-238 < h < -3.999999999999988e-310

   1. Initial program 39.5%

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

   4. Applied rewrites 86.3%

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

   if -3.999999999999988e-310 < h

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

   6. Applied rewrites 70.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 72.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 80.5

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

   10. Applied rewrites 80.5%

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

Alternative 2: 49.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-161} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+135}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -4e-220)
     (* (sqrt (pow (* l h) -1.0)) d)
     (if (or (<= t_0 2e-161) (not (<= t_0 5e+135)))
       (fabs (/ d (sqrt (* l h))))
       (sqrt (* (/ (/ d l) h) d))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -4e-220) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else if ((t_0 <= 2e-161) || !(t_0 <= 5e+135)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = sqrt((((d / l) / h) * d));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-4d-220)) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else if ((t_0 <= 2d-161) .or. (.not. (t_0 <= 5d+135))) then
        tmp = abs((d / sqrt((l * h))))
    else
        tmp = sqrt((((d / l) / h) * d))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -4e-220) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else if ((t_0 <= 2e-161) || !(t_0 <= 5e+135)) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = Math.sqrt((((d / l) / h) * d));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -4e-220:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	elif (t_0 <= 2e-161) or not (t_0 <= 5e+135):
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = math.sqrt((((d / l) / h) * d))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -4e-220)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	elseif ((t_0 <= 2e-161) || !(t_0 <= 5e+135))
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = sqrt(Float64(Float64(Float64(d / l) / h) * d));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -4e-220)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	elseif ((t_0 <= 2e-161) || ~((t_0 <= 5e+135)))
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = sqrt((((d / l) / h) * d));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-220], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[Or[LessEqual[t$95$0, 2e-161], N[Not[LessEqual[t$95$0, 5e+135]], $MachinePrecision]], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -3.99999999999999997e-220

   1. Initial program 83.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 12.3

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

   5. Applied rewrites 12.3%

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

   if -3.99999999999999997e-220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.00000000000000006e-161 or 5.00000000000000029e135 < (*.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 39.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 32.2

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

   5. Applied rewrites 32.2%

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

   7. Applied rewrites 59.9%

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

   if 2.00000000000000006e-161 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000029e135

   1. Initial program 99.5%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   7. Applied rewrites 97.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{-161} \lor \neg \left(\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+135}\right):\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right) \cdot -0.125, \frac{M\_m \cdot M\_m}{\ell}, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -5e-15)
     (*
      (* (fma (* (* (/ (/ (* D D) d) d) h) -0.125) (/ (* M_m M_m) l) 1.0) t_1)
      t_2)
     (if (<= t_0 5e+270) (* t_2 t_1) (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-15) {
		tmp = (fma((((((D * D) / d) / d) * h) * -0.125), ((M_m * M_m) / l), 1.0) * t_1) * t_2;
	} else if (t_0 <= 5e+270) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -5e-15)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D * D) / d) / d) * h) * -0.125), Float64(Float64(M_m * M_m) / l), 1.0) * t_1) * t_2);
	elseif (t_0 <= 5e+270)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-15], N[(N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+270], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999999e-15

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. associate-*r/ N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

   7. Applied rewrites 75.0%

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

   if -4.99999999999999999e-15 < (*.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)))) < 4.99999999999999976e270

   1. Initial program 89.8%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.9

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

   5. Applied rewrites 41.9%

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

   7. Applied rewrites 86.7%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right) \cdot -0.125, \frac{M \cdot M}{\ell}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right) \cdot -0.125\right) \cdot \frac{M\_m \cdot M\_m}{\ell}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d h)))
        (t_2 (sqrt (/ d l))))
   (if (<= t_0 -5e-15)
     (* (* (* (* (* (/ (/ (* D D) d) d) h) -0.125) (/ (* M_m M_m) l)) t_1) t_2)
     (if (<= t_0 5e+270) (* t_2 t_1) (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / h));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-15) {
		tmp = (((((((D * D) / d) / d) * h) * -0.125) * ((M_m * M_m) / l)) * t_1) * t_2;
	} else if (t_0 <= 5e+270) {
		tmp = t_2 * t_1;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = sqrt((d / h))
    t_2 = sqrt((d / l))
    if (t_0 <= (-5d-15)) then
        tmp = (((((((d_1 * d_1) / d) / d) * h) * (-0.125d0)) * ((m_m * m_m) / l)) * t_1) * t_2
    else if (t_0 <= 5d+270) then
        tmp = t_2 * t_1
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d / h));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_0 <= -5e-15) {
		tmp = (((((((D * D) / d) / d) * h) * -0.125) * ((M_m * M_m) / l)) * t_1) * t_2;
	} else if (t_0 <= 5e+270) {
		tmp = t_2 * t_1;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d / h))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_0 <= -5e-15:
		tmp = (((((((D * D) / d) / d) * h) * -0.125) * ((M_m * M_m) / l)) * t_1) * t_2
	elif t_0 <= 5e+270:
		tmp = t_2 * t_1
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_0 <= -5e-15)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(D * D) / d) / d) * h) * -0.125) * Float64(Float64(M_m * M_m) / l)) * t_1) * t_2);
	elseif (t_0 <= 5e+270)
		tmp = Float64(t_2 * t_1);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d / h));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_0 <= -5e-15)
		tmp = (((((((D * D) / d) / d) * h) * -0.125) * ((M_m * M_m) / l)) * t_1) * t_2;
	elseif (t_0 <= 5e+270)
		tmp = t_2 * t_1;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-15], N[(N[(N[(N[(N[(N[(N[(N[(D * D), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision] * h), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(M$95$m * M$95$m), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 5e+270], N[(t$95$2 * t$95$1), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999999e-15

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l/ N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 73.6%

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

   if -4.99999999999999999e-15 < (*.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)))) < 4.99999999999999976e270

   1. Initial program 89.8%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.9

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

   5. Applied rewrites 41.9%

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

   7. Applied rewrites 86.7%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(\left(\left(\frac{\frac{D \cdot D}{d}}{d} \cdot h\right) \cdot -0.125\right) \cdot \frac{M \cdot M}{\ell}\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 0.0)
     (* (sqrt (pow (* l h) -1.0)) d)
     (if (<= t_0 5e+270)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (fabs (/ d (sqrt (* l h))))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else if (t_0 <= 5e+270) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else if (t_0 <= 5d+270) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else if (t_0 <= 5e+270) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	elif t_0 <= 5e+270:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	elseif (t_0 <= 5e+270)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	elseif (t_0 <= 5e+270)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], If[LessEqual[t$95$0, 5e+270], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.3%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 18.2

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

   5. Applied rewrites 18.2%

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

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 98.9%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 41.5

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

   5. Applied rewrites 41.5%

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

   7. Applied rewrites 98.9%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{t\_0}{\ell}, \frac{t\_0 \cdot -0.125}{{h}^{-1}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D d))))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        5e+270)
     (*
      (* (fma (/ t_0 l) (/ (* t_0 -0.125) (pow h -1.0)) 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (fabs (/ d (sqrt (* l h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / d);
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (fma((t_0 / l), ((t_0 * -0.125) / pow(h, -1.0)), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+270)
		tmp = Float64(Float64(fma(Float64(t_0 / l), Float64(Float64(t_0 * -0.125) / (h ^ -1.0)), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+270], N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * N[(N[(t$95$0 * -0.125), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 88.0%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{D}{d} \cdot M\_m\\ t_1 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\frac{\frac{t\_0}{\ell} \cdot \left(0.125 \cdot t\_0\right)}{\frac{-1}{h}} \cdot t\_1 + t\_1\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (/ D d) M_m)) (t_1 (sqrt (/ d h))))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        5e+270)
     (*
      (+ (* (/ (* (/ t_0 l) (* 0.125 t_0)) (/ -1.0 h)) t_1) t_1)
      (sqrt (/ d l)))
     (fabs (/ d (sqrt (* l h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (D / d) * M_m;
	double t_1 = sqrt((d / h));
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (((((t_0 / l) * (0.125 * t_0)) / (-1.0 / h)) * t_1) + t_1) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_1 / d) * m_m
    t_1 = sqrt((d / h))
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= 5d+270) then
        tmp = (((((t_0 / l) * (0.125d0 * t_0)) / ((-1.0d0) / h)) * t_1) + t_1) * sqrt((d / l))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (D / d) * M_m;
	double t_1 = Math.sqrt((d / h));
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (((((t_0 / l) * (0.125 * t_0)) / (-1.0 / h)) * t_1) + t_1) * Math.sqrt((d / l));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (D / d) * M_m
	t_1 = math.sqrt((d / h))
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270:
		tmp = (((((t_0 / l) * (0.125 * t_0)) / (-1.0 / h)) * t_1) + t_1) * math.sqrt((d / l))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(D / d) * M_m)
	t_1 = sqrt(Float64(d / h))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+270)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(t_0 / l) * Float64(0.125 * t_0)) / Float64(-1.0 / h)) * t_1) + t_1) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (D / d) * M_m;
	t_1 = sqrt((d / h));
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= 5e+270)
		tmp = (((((t_0 / l) * (0.125 * t_0)) / (-1.0 / h)) * t_1) + t_1) * sqrt((d / l));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+270], N[(N[(N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * N[(0.125 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$1), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 84.6%

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

   8. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\left(\frac{h \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{d}{h}} + \sqrt{\frac{d}{h}}\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lift-/.f64 N/A

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

      13. remove-double-div N/A

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

      14. unpow-1 N/A

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

      15. lift-pow.f64 N/A

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

      16. div-inv N/A

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

      17. associate-*r/ N/A

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

   10. Applied rewrites 87.2%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\frac{\frac{\frac{D}{d} \cdot M}{\ell} \cdot \left(0.125 \cdot \left(\frac{D}{d} \cdot M\right)\right)}{\frac{-1}{h}} \cdot \sqrt{\frac{d}{h}} + \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(0.5 \cdot \frac{D}{d}\right) \cdot M\_m\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot t\_0, t\_0, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (/ D d)) M_m)))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        5e+270)
     (*
      (* (fma (* (* (/ h l) -0.5) t_0) t_0 1.0) (sqrt (/ d h)))
      (sqrt (/ d l)))
     (fabs (/ d (sqrt (* l h)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (0.5 * (D / d)) * M_m;
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (fma((((h / l) * -0.5) * t_0), t_0, 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(0.5 * Float64(D / d)) * M_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+270)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(h / l) * -0.5) * t_0), t_0, 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+270], N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. unpow2 N/A

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

      16. associate-*r* N/A

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

   6. Applied rewrites 87.3%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\frac{h}{\ell} \cdot -0.5\right) \cdot \left(\left(0.5 \cdot \frac{D}{d}\right) \cdot M\right), \left(0.5 \cdot \frac{D}{d}\right) \cdot M, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      -4e-220)
   (* (sqrt (pow (* l h) -1.0)) d)
   (fabs (/ d (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-220) {
		tmp = sqrt(pow((l * h), -1.0)) * d;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-4d-220)) then
        tmp = sqrt(((l * h) ** (-1.0d0))) * d
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-220) {
		tmp = Math.sqrt(Math.pow((l * h), -1.0)) * d;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-220:
		tmp = math.sqrt(math.pow((l * h), -1.0)) * d
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -4e-220)
		tmp = Float64(sqrt((Float64(l * h) ^ -1.0)) * d);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -4e-220)
		tmp = sqrt(((l * h) ^ -1.0)) * d;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-220], N[(N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -3.99999999999999997e-220

   1. Initial program 83.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 12.3

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

   5. Applied rewrites 12.3%

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

   if -3.99999999999999997e-220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 59.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 66.4%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{{\left(\ell \cdot h\right)}^{-1}} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M\_m \cdot \frac{D}{d}}{\ell}, \frac{D}{d} \cdot \left(M\_m \cdot \left(h \cdot -0.125\right)\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      5e+270)
   (*
    (*
     (fma (/ (* M_m (/ D d)) l) (* (/ D d) (* M_m (* h -0.125))) 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (fabs (/ d (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (fma(((M_m * (D / d)) / l), ((D / d) * (M_m * (h * -0.125))), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+270)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * Float64(D / d)) / l), Float64(Float64(D / d) * Float64(M_m * Float64(h * -0.125))), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+270], N[(N[(N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(D / d), $MachinePrecision] * N[(M$95$m * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 88.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow-1 N/A

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

      10. associate-/r/ N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 84.7

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

   10. Applied rewrites 84.7%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{D}{d} \cdot \left(M \cdot \left(h \cdot -0.125\right)\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M\_m \cdot \frac{D}{d}}{\ell}, M\_m \cdot \left(\frac{D}{d} \cdot \left(h \cdot -0.125\right)\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      5e+270)
   (*
    (*
     (fma (/ (* M_m (/ D d)) l) (* M_m (* (/ D d) (* h -0.125))) 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (fabs (/ d (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (fma(((M_m * (D / d)) / l), (M_m * ((D / d) * (h * -0.125))), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+270)
		tmp = Float64(Float64(fma(Float64(Float64(M_m * Float64(D / d)) / l), Float64(M_m * Float64(Float64(D / d) * Float64(h * -0.125))), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+270], N[(N[(N[(N[(N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(M$95$m * N[(N[(D / d), $MachinePrecision] * N[(h * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 88.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow-1 N/A

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

      9. associate-/r/ N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 86.9

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

   10. Applied rewrites 86.9%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, M \cdot \left(\frac{D}{d} \cdot \left(h \cdot -0.125\right)\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(M\_m, \frac{\frac{D}{d}}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\frac{D}{d} \cdot M\_m\right)\right)\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
      5e+270)
   (*
    (*
     (fma M_m (* (/ (/ D d) l) (* h (* -0.125 (* (/ D d) M_m)))) 1.0)
     (sqrt (/ d h)))
    (sqrt (/ d l)))
   (fabs (/ d (sqrt (* l h))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+270) {
		tmp = (fma(M_m, (((D / d) / l) * (h * (-0.125 * ((D / d) * M_m)))), 1.0) * sqrt((d / h))) * sqrt((d / l));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+270)
		tmp = Float64(Float64(fma(M_m, Float64(Float64(Float64(D / d) / l) * Float64(h * Float64(-0.125 * Float64(Float64(D / d) * M_m)))), 1.0) * sqrt(Float64(d / h))) * sqrt(Float64(d / l)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+270], N[(N[(N[(M$95$m * N[(N[(N[(D / d), $MachinePrecision] / l), $MachinePrecision] * N[(h * N[(-0.125 * N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999976e270

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 84.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 88.0%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 86.5%

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

   if 4.99999999999999976e270 < (*.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 22.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 26.5

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

   5. Applied rewrites 26.5%

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

   7. Applied rewrites 53.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\left(\mathsf{fma}\left(M, \frac{\frac{D}{d}}{\ell} \cdot \left(h \cdot \left(-0.125 \cdot \left(\frac{D}{d} \cdot M\right)\right)\right), 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M\_m \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<=
        (*
         (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
         (-
          1.0
          (* (* (pow 2.0 -1.0) (pow (/ (* M_m D) (* 2.0 d)) 2.0)) (/ h l))))
        -4e-220)
     t_0
     (fabs t_0))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (((pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-220) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m_m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-4d-220)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-220) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M_m * D) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-220:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M_m * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -4e-220)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M_m * D) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -4e-220)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-220], t$95$0, N[Abs[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -3.99999999999999997e-220

   1. Initial program 83.4%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 12.3

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

   5. Applied rewrites 12.3%

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

   7. Applied rewrites 11.3%

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

   if -3.99999999999999997e-220 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 59.6%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 66.4%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d}\\ t_1 := \sqrt{\frac{d}{h}}\\ t_2 := \sqrt{-d}\\ \mathbf{if}\;h \leq -2.05 \cdot 10^{-171}:\\ \;\;\;\;\left(\frac{h \cdot \left(-0.125 \cdot {t\_0}^{2}\right)}{\ell} \cdot t\_1 + t\_1\right) \cdot \frac{t\_2}{\sqrt{-\ell}}\\ \mathbf{elif}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M\_m} \cdot \frac{2}{D}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot t\_2}{\sqrt{-h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{t\_0}{\ell}, \frac{t\_0 \cdot -0.125}{{h}^{-1}}, 1\right) \cdot t\_1\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D d))) (t_1 (sqrt (/ d h))) (t_2 (sqrt (- d))))
   (if (<= h -2.05e-171)
     (*
      (+ (* (/ (* h (* -0.125 (pow t_0 2.0))) l) t_1) t_1)
      (/ t_2 (sqrt (- l))))
     (if (<= h -4e-310)
       (/
        (*
         (*
          (fma (* -0.5 (/ h l)) (pow (* (/ d M_m) (/ 2.0 D)) -2.0) 1.0)
          (sqrt (/ d l)))
         t_2)
        (sqrt (- h)))
       (*
        (* (fma (/ t_0 l) (/ (* t_0 -0.125) (pow h -1.0)) 1.0) t_1)
        (/ (sqrt d) (sqrt l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / d);
	double t_1 = sqrt((d / h));
	double t_2 = sqrt(-d);
	double tmp;
	if (h <= -2.05e-171) {
		tmp = ((((h * (-0.125 * pow(t_0, 2.0))) / l) * t_1) + t_1) * (t_2 / sqrt(-l));
	} else if (h <= -4e-310) {
		tmp = ((fma((-0.5 * (h / l)), pow(((d / M_m) * (2.0 / D)), -2.0), 1.0) * sqrt((d / l))) * t_2) / sqrt(-h);
	} else {
		tmp = (fma((t_0 / l), ((t_0 * -0.125) / pow(h, -1.0)), 1.0) * t_1) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d))
	t_1 = sqrt(Float64(d / h))
	t_2 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -2.05e-171)
		tmp = Float64(Float64(Float64(Float64(Float64(h * Float64(-0.125 * (t_0 ^ 2.0))) / l) * t_1) + t_1) * Float64(t_2 / sqrt(Float64(-l))));
	elseif (h <= -4e-310)
		tmp = Float64(Float64(Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M_m) * Float64(2.0 / D)) ^ -2.0), 1.0) * sqrt(Float64(d / l))) * t_2) / sqrt(Float64(-h)));
	else
		tmp = Float64(Float64(fma(Float64(t_0 / l), Float64(Float64(t_0 * -0.125) / (h ^ -1.0)), 1.0) * t_1) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -2.05e-171], N[(N[(N[(N[(N[(h * N[(-0.125 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$1), $MachinePrecision] * N[(t$95$2 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -4e-310], N[(N[(N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M$95$m), $MachinePrecision] * N[(2.0 / D), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * N[(N[(t$95$0 * -0.125), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -2.05e-171

   1. Initial program 69.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.0%

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

   6. Applied rewrites 72.3%

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

   8. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(\frac{h \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{d}{h}} + \sqrt{\frac{d}{h}}\right)} \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-neg.f64 79.0

          \[\leadsto \left(\frac{h \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{d}{h}} + \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{-d}}{\sqrt{\color{blue}{-\ell}}} \]

   10. Applied rewrites 79.0%

       \[\leadsto \left(\frac{h \cdot \left(-0.125 \cdot {\left(M \cdot \frac{D}{d}\right)}^{2}\right)}{\ell} \cdot \sqrt{\frac{d}{h}} + \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \]

   if -2.05e-171 < h < -3.999999999999988e-310

   1. Initial program 52.4%

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

   4. Applied rewrites 82.2%

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

   if -3.999999999999988e-310 < h

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

   6. Applied rewrites 70.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 72.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 80.5

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

   10. Applied rewrites 80.5%

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

Alternative 15: 77.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \frac{D}{d}\\ \mathbf{if}\;h \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{D}{d} \cdot M\_m\right)}^{2} \cdot 0.25, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{t\_0}{\ell}, \frac{t\_0 \cdot -0.125}{{h}^{-1}}, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \frac{\sqrt{d}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (/ D d))))
   (if (<= h -4e-310)
     (*
      (/
       (*
        (fma (* (pow (* (/ D d) M_m) 2.0) 0.25) (* (/ h l) -0.5) 1.0)
        (sqrt (- d)))
       (sqrt (- h)))
      (sqrt (/ d l)))
     (*
      (* (fma (/ t_0 l) (/ (* t_0 -0.125) (pow h -1.0)) 1.0) (sqrt (/ d h)))
      (/ (sqrt d) (sqrt l))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D / d);
	double tmp;
	if (h <= -4e-310) {
		tmp = ((fma((pow(((D / d) * M_m), 2.0) * 0.25), ((h / l) * -0.5), 1.0) * sqrt(-d)) / sqrt(-h)) * sqrt((d / l));
	} else {
		tmp = (fma((t_0 / l), ((t_0 * -0.125) / pow(h, -1.0)), 1.0) * sqrt((d / h))) * (sqrt(d) / sqrt(l));
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D / d))
	tmp = 0.0
	if (h <= -4e-310)
		tmp = Float64(Float64(Float64(fma(Float64((Float64(Float64(D / d) * M_m) ^ 2.0) * 0.25), Float64(Float64(h / l) * -0.5), 1.0) * sqrt(Float64(-d))) / sqrt(Float64(-h))) * sqrt(Float64(d / l)));
	else
		tmp = Float64(Float64(fma(Float64(t_0 / l), Float64(Float64(t_0 * -0.125) / (h ^ -1.0)), 1.0) * sqrt(Float64(d / h))) * Float64(sqrt(d) / sqrt(l)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -4e-310], N[(N[(N[(N[(N[(N[Power[N[(N[(D / d), $MachinePrecision] * M$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 / l), $MachinePrecision] * N[(N[(t$95$0 * -0.125), $MachinePrecision] / N[Power[h, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -3.999999999999988e-310

   1. Initial program 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 67.1%

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

   6. Applied rewrites 75.9%

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

   if -3.999999999999988e-310 < h

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 70.7%

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

   6. Applied rewrites 70.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. div-inv N/A

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

      16. times-frac N/A

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

   8. Applied rewrites 72.6%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{M \cdot \frac{D}{d}}{\ell}, \frac{\left(M \cdot \frac{D}{d}\right) \cdot -0.125}{{h}^{-1}}, 1\right)} \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}} \]
   9. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 80.5

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

   10. Applied rewrites 80.5%

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

Alternative 16: 26.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D) :precision binary64 (/ d (sqrt (* l h))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	return d / sqrt((l * h));
}

Fortran

M_m = abs(m)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m_m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	return d / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	return d / math.sqrt((l * h))

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp = code(d, h, l, M_m, D)
	tmp = d / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 27.4

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

5. Applied rewrites 27.4%

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

7. Applied rewrites 27.3%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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)))))