Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.6% → 79.8%

Time: 15.8s

Alternatives: 19

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: 65.6% 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: 79.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)}\\ t_1 := \left(t\_0 \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{D\_m}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \left(0.25 \cdot D\_m\right)\right) \cdot h\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(D\_m \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (pow (/ d h) (pow 2.0 -1.0)))
        (t_1
         (*
          (* t_0 (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 4e+250)
     (*
      (* t_0 (sqrt (/ d l)))
      (- 1.0 (* (/ (* (* (/ D_m d) M) 0.5) l) (* (* (/ M d) (* 0.25 D_m)) h))))
     (if (<= t_1 INFINITY)
       (fabs (/ d (sqrt (* l h))))
       (*
        (/ (fma (/ (* (pow (* D_m M) 2.0) -0.125) d) (/ h l) d) h)
        (sqrt (/ h l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = pow((d / h), pow(2.0, -1.0));
	double t_1 = (t_0 * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 4e+250) {
		tmp = (t_0 * sqrt((d / l))) * (1.0 - (((((D_m / d) * M) * 0.5) / l) * (((M / d) * (0.25 * D_m)) * h)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = (fma(((pow((D_m * M), 2.0) * -0.125) / d), (h / l), d) / h) * sqrt((h / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(d / h) ^ (2.0 ^ -1.0)
	t_1 = Float64(Float64(t_0 * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 4e+250)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / l))) * Float64(1.0 - Float64(Float64(Float64(Float64(Float64(D_m / d) * M) * 0.5) / l) * Float64(Float64(Float64(M / d) * Float64(0.25 * D_m)) * h))));
	elseif (t_1 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64((Float64(D_m * M) ^ 2.0) * -0.125) / d), Float64(h / l), d) / h) * sqrt(Float64(h / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+250], N[(N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision] * 0.5), $MachinePrecision] / l), $MachinePrecision] * N[(N[(N[(M / d), $MachinePrecision] * N[(0.25 * D$95$m), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[(N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision] / h), $MachinePrecision] * N[Sqrt[N[(h / l), $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)))) < 3.9999999999999997e250

   1. Initial program 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. div-inv N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 91.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow-1 N/A

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

      5. remove-double-div N/A

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

      6. lower-*.f64 91.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.0

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 91.0

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

   6. Applied rewrites 91.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 91.0

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

      3. lift-pow.f64 N/A

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

      4. unpow1/2 N/A

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

      5. lower-sqrt.f64 91.0

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

   8. Applied rewrites 91.0%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 15.6%

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

   7. Applied rewrites 18.4%

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

   9. Applied rewrites 62.1%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\left(\frac{D}{d} \cdot M\right) \cdot 0.5}{\ell} \cdot \left(\left(\frac{M}{d} \cdot \left(0.25 \cdot D\right)\right) \cdot h\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := -0.125 \cdot \left(D\_m \cdot D\_m\right)\\ t_1 := \frac{M \cdot M}{d}\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \sqrt{\frac{d}{\ell}}\\ t_4 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+250}:\\ \;\;\;\;\left(t\_3 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\frac{h}{\ell} \cdot -0.25\right) \cdot M\right) \cdot \frac{D\_m}{d}\right) \cdot \left(M \cdot 0.5\right), \frac{D\_m}{d}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{elif}\;t\_2 \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{t\_4 \cdot \mathsf{fma}\left(\frac{h}{\ell}, t\_0 \cdot t\_1, d\right)}{h}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{t\_3}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4 \cdot \frac{\mathsf{fma}\left(t\_0, h \cdot t\_1, \ell \cdot d\right)}{\ell}}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (* -0.125 (* D_m D_m)))
        (t_1 (/ (* M M) d))
        (t_2
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (sqrt (/ d l)))
        (t_4 (sqrt (/ h l))))
   (if (<= t_2 -1e+250)
     (*
      (* t_3 (sqrt (/ d h)))
      (* (* (* h (* M (/ M l))) -0.125) (/ (/ (* D_m D_m) d) d)))
     (if (<= t_2 -1e+131)
       (*
        (fma (* (* (* (* (/ h l) -0.25) M) (/ D_m d)) (* M 0.5)) (/ D_m d) 1.0)
        (sqrt (* (/ (/ d l) h) d)))
       (if (<= t_2 2.5e-112)
         (/ (* t_4 (fma (/ h l) (* t_0 t_1) d)) h)
         (if (<= t_2 4e+250)
           (/ t_3 (sqrt (/ h d)))
           (if (<= t_2 INFINITY)
             (fabs (/ d (sqrt (* l h))))
             (/ (* t_4 (/ (fma t_0 (* h t_1) (* l d)) l)) h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = -0.125 * (D_m * D_m);
	double t_1 = (M * M) / d;
	double t_2 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = sqrt((d / l));
	double t_4 = sqrt((h / l));
	double tmp;
	if (t_2 <= -1e+250) {
		tmp = (t_3 * sqrt((d / h))) * (((h * (M * (M / l))) * -0.125) * (((D_m * D_m) / d) / d));
	} else if (t_2 <= -1e+131) {
		tmp = fma((((((h / l) * -0.25) * M) * (D_m / d)) * (M * 0.5)), (D_m / d), 1.0) * sqrt((((d / l) / h) * d));
	} else if (t_2 <= 2.5e-112) {
		tmp = (t_4 * fma((h / l), (t_0 * t_1), d)) / h;
	} else if (t_2 <= 4e+250) {
		tmp = t_3 / sqrt((h / d));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = (t_4 * (fma(t_0, (h * t_1), (l * d)) / l)) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(-0.125 * Float64(D_m * D_m))
	t_1 = Float64(Float64(M * M) / d)
	t_2 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = sqrt(Float64(d / l))
	t_4 = sqrt(Float64(h / l))
	tmp = 0.0
	if (t_2 <= -1e+250)
		tmp = Float64(Float64(t_3 * sqrt(Float64(d / h))) * Float64(Float64(Float64(h * Float64(M * Float64(M / l))) * -0.125) * Float64(Float64(Float64(D_m * D_m) / d) / d)));
	elseif (t_2 <= -1e+131)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(h / l) * -0.25) * M) * Float64(D_m / d)) * Float64(M * 0.5)), Float64(D_m / d), 1.0) * sqrt(Float64(Float64(Float64(d / l) / h) * d)));
	elseif (t_2 <= 2.5e-112)
		tmp = Float64(Float64(t_4 * fma(Float64(h / l), Float64(t_0 * t_1), d)) / h);
	elseif (t_2 <= 4e+250)
		tmp = Float64(t_3 / sqrt(Float64(h / d)));
	elseif (t_2 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(t_4 * Float64(fma(t_0, Float64(h * t_1), Float64(l * d)) / l)) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1e+250], N[(N[(t$95$3 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e+131], N[(N[(N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.25), $MachinePrecision] * M), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.5e-112], N[(N[(t$95$4 * N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * t$95$1), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$2, 4e+250], N[(t$95$3 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$4 * N[(N[(t$95$0 * N[(h * t$95$1), $MachinePrecision] + N[(l * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999992e249

   1. Initial program 86.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 41.8

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

   4. Applied rewrites 41.8%

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

   5. Taylor expanded in d around 0

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

      1. metadata-eval N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

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

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 39.3%

      \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.3

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)} \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 39.3

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \]

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 39.3

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

      17. lower-sqrt.f64 83.4

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

   9. Applied rewrites 83.4%

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

   if -9.9999999999999992e249 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -9.9999999999999991e130

   1. Initial program 98.8%

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

   4. Applied rewrites 82.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   6. Applied rewrites 82.9%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 67.6%

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

   if -9.9999999999999991e130 < (*.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)))) < 2.50000000000000022e-112

   1. Initial program 67.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 69.8%

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

   9. Applied rewrites 79.5%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in d around 0

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

   12. Applied rewrites 77.2%

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

   if 2.50000000000000022e-112 < (*.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)))) < 3.9999999999999997e250

   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 99.6%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 15.6%

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

   7. Applied rewrites 18.4%

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

   9. Applied rewrites 62.1%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 39.5%

       \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{d}, \ell \cdot d\right)}{\ell}}{h} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.9%

   \[\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 -1 \cdot 10^{+250}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)\\ \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 -1 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(\frac{h}{\ell} \cdot -0.25\right) \cdot M\right) \cdot \frac{D}{d}\right) \cdot \left(M \cdot 0.5\right), \frac{D}{d}, 1\right) \cdot \sqrt{\frac{\frac{d}{\ell}}{h} \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 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}, d\right)}{h}\\ \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 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{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 \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{d}, \ell \cdot d\right)}{\ell}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D\_m \cdot D\_m}{d}}{d}\right)\\ \mathbf{elif}\;t\_1 \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot D\_m\right) \cdot \frac{D\_m}{d}, \frac{h}{\ell} \cdot t\_2, t\_2 \cdot d\right)}{h}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{t\_0}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot \frac{\mathsf{fma}\left(-0.125 \cdot \left(D\_m \cdot D\_m\right), h \cdot \frac{M \cdot M}{d}, \ell \cdot d\right)}{\ell}}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ h l))))
   (if (<= t_1 (- INFINITY))
     (*
      (* t_0 (sqrt (/ d h)))
      (* (* (* h (* M (/ M l))) -0.125) (/ (/ (* D_m D_m) d) d)))
     (if (<= t_1 2.5e-112)
       (/
        (fma
         (* (* (* -0.125 (* M M)) D_m) (/ D_m d))
         (* (/ h l) t_2)
         (* t_2 d))
        h)
       (if (<= t_1 4e+250)
         (/ t_0 (sqrt (/ h d)))
         (if (<= t_1 INFINITY)
           (fabs (/ d (sqrt (* l h))))
           (/
            (*
             t_2
             (/ (fma (* -0.125 (* D_m D_m)) (* h (/ (* M M) d)) (* l d)) l))
            h)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((h / l));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_0 * sqrt((d / h))) * (((h * (M * (M / l))) * -0.125) * (((D_m * D_m) / d) / d));
	} else if (t_1 <= 2.5e-112) {
		tmp = fma((((-0.125 * (M * M)) * D_m) * (D_m / d)), ((h / l) * t_2), (t_2 * d)) / h;
	} else if (t_1 <= 4e+250) {
		tmp = t_0 / sqrt((h / d));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = (t_2 * (fma((-0.125 * (D_m * D_m)), (h * ((M * M) / d)), (l * d)) / l)) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(h / l))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * Float64(Float64(Float64(h * Float64(M * Float64(M / l))) * -0.125) * Float64(Float64(Float64(D_m * D_m) / d) / d)));
	elseif (t_1 <= 2.5e-112)
		tmp = Float64(fma(Float64(Float64(Float64(-0.125 * Float64(M * M)) * D_m) * Float64(D_m / d)), Float64(Float64(h / l) * t_2), Float64(t_2 * d)) / h);
	elseif (t_1 <= 4e+250)
		tmp = Float64(t_0 / sqrt(Float64(h / d)));
	elseif (t_1 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(t_2 * Float64(fma(Float64(-0.125 * Float64(D_m * D_m)), Float64(h * Float64(Float64(M * M) / d)), Float64(l * d)) / l)) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(h * N[(M * N[(M / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.5e-112], N[(N[(N[(N[(N[(-0.125 * N[(M * M), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$2 * d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 4e+250], N[(t$95$0 / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$2 * N[(N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(h * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + N[(l * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 86.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 42.3

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

   4. Applied rewrites 42.3%

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

   5. Taylor expanded in d around 0

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

      1. metadata-eval N/A

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

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

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

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

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 39.8%

      \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right)} \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \]

      4. lift-/.f64 N/A

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

      5. metadata-eval 39.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot \frac{\sqrt{-d}}{\sqrt{-h}}\right) \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right) \]

      6. lift-pow.f64 N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 39.8

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-undiv N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

      17. lower-sqrt.f64 83.2

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

   9. Applied rewrites 83.2%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.50000000000000022e-112

   1. Initial program 73.8%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 38.0%

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

   7. Applied rewrites 58.7%

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

   9. Applied rewrites 68.6%

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

   if 2.50000000000000022e-112 < (*.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)))) < 3.9999999999999997e250

   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 99.6%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 15.6%

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

   7. Applied rewrites 18.4%

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

   9. Applied rewrites 62.1%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 39.5%

       \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{d}, \ell \cdot d\right)}{\ell}}{h} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -\infty:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(\left(\left(h \cdot \left(M \cdot \frac{M}{\ell}\right)\right) \cdot -0.125\right) \cdot \frac{\frac{D \cdot D}{d}}{d}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(-0.125 \cdot \left(M \cdot M\right)\right) \cdot D\right) \cdot \frac{D}{d}, \frac{h}{\ell} \cdot \sqrt{\frac{h}{\ell}}, \sqrt{\frac{h}{\ell}} \cdot d\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{d}, \ell \cdot d\right)}{\ell}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{{\left(D\_m \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;t\_0 \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (*
          (/ (fma (/ (* (pow (* D_m M) 2.0) -0.125) d) (/ h l) d) h)
          (sqrt (/ h l)))))
   (if (<= t_0 2.5e-112)
     t_1
     (if (<= t_0 4e+250)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (if (<= t_0 INFINITY) (fabs (/ d (sqrt (* l h)))) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (fma(((pow((D_m * M), 2.0) * -0.125) / d), (h / l), d) / h) * sqrt((h / l));
	double tmp;
	if (t_0 <= 2.5e-112) {
		tmp = t_1;
	} else if (t_0 <= 4e+250) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(fma(Float64(Float64((Float64(D_m * M) ^ 2.0) * -0.125) / d), Float64(h / l), d) / h) * sqrt(Float64(h / l)))
	tmp = 0.0
	if (t_0 <= 2.5e-112)
		tmp = t_1;
	elseif (t_0 <= 4e+250)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (t_0 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision] / h), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.5e-112], t$95$1, If[LessEqual[t$95$0, 4e+250], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.50000000000000022e-112 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 61.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 43.7%

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

   7. Applied rewrites 53.5%

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

   9. Applied rewrites 78.3%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.1%

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

   if 2.50000000000000022e-112 < (*.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)))) < 3.9999999999999997e250

   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 99.6%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := -0.125 \cdot \left(D\_m \cdot D\_m\right)\\ t_2 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_3 := \frac{M \cdot M}{d}\\ \mathbf{if}\;t\_2 \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\frac{h}{\ell}, t\_1 \cdot t\_3, d\right)}{h}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{\mathsf{fma}\left(t\_1, h \cdot t\_3, \ell \cdot d\right)}{\ell}}{h}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1 (* -0.125 (* D_m D_m)))
        (t_2
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_3 (/ (* M M) d)))
   (if (<= t_2 2.5e-112)
     (/ (* t_0 (fma (/ h l) (* t_1 t_3) d)) h)
     (if (<= t_2 4e+250)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (if (<= t_2 INFINITY)
         (fabs (/ d (sqrt (* l h))))
         (/ (* t_0 (/ (fma t_1 (* h t_3) (* l d)) l)) h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = -0.125 * (D_m * D_m);
	double t_2 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_3 = (M * M) / d;
	double tmp;
	if (t_2 <= 2.5e-112) {
		tmp = (t_0 * fma((h / l), (t_1 * t_3), d)) / h;
	} else if (t_2 <= 4e+250) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = (t_0 * (fma(t_1, (h * t_3), (l * d)) / l)) / h;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(-0.125 * Float64(D_m * D_m))
	t_2 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_3 = Float64(Float64(M * M) / d)
	tmp = 0.0
	if (t_2 <= 2.5e-112)
		tmp = Float64(Float64(t_0 * fma(Float64(h / l), Float64(t_1 * t_3), d)) / h);
	elseif (t_2 <= 4e+250)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (t_2 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(t_0 * Float64(fma(t_1, Float64(h * t_3), Float64(l * d)) / l)) / h);
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[t$95$2, 2.5e-112], N[(N[(t$95$0 * N[(N[(h / l), $MachinePrecision] * N[(t$95$1 * t$95$3), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$2, 4e+250], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(t$95$0 * N[(N[(t$95$1 * N[(h * t$95$3), $MachinePrecision] + N[(l * d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.50000000000000022e-112

   1. Initial program 82.6%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 53.3%

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

   7. Applied rewrites 65.5%

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

   9. Applied rewrites 83.8%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in d around 0

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

   12. Applied rewrites 76.1%

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

   if 2.50000000000000022e-112 < (*.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)))) < 3.9999999999999997e250

   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 99.6%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 15.6%

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

   7. Applied rewrites 18.4%

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

   9. Applied rewrites 62.1%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in l around 0

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

   12. Applied rewrites 39.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}, d\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \frac{\mathsf{fma}\left(-0.125 \cdot \left(D \cdot D\right), h \cdot \frac{M \cdot M}{d}, \ell \cdot d\right)}{\ell}}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \left(-0.125 \cdot \left(D\_m \cdot D\_m\right)\right) \cdot \frac{M \cdot M}{d}, d\right)}{h}\\ \mathbf{if}\;t\_0 \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (/
          (*
           (sqrt (/ h l))
           (fma (/ h l) (* (* -0.125 (* D_m D_m)) (/ (* M M) d)) d))
          h)))
   (if (<= t_0 2.5e-112)
     t_1
     (if (<= t_0 4e+250)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (if (<= t_0 INFINITY) (fabs (/ d (sqrt (* l h)))) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (sqrt((h / l)) * fma((h / l), ((-0.125 * (D_m * D_m)) * ((M * M) / d)), d)) / h;
	double tmp;
	if (t_0 <= 2.5e-112) {
		tmp = t_1;
	} else if (t_0 <= 4e+250) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(h / l), Float64(Float64(-0.125 * Float64(D_m * D_m)) * Float64(Float64(M * M) / d)), d)) / h)
	tmp = 0.0
	if (t_0 <= 2.5e-112)
		tmp = t_1;
	elseif (t_0 <= 4e+250)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (t_0 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[(-0.125 * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(M * M), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$0, 2.5e-112], t$95$1, If[LessEqual[t$95$0, 4e+250], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 2.50000000000000022e-112 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 61.5%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 43.7%

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

   7. Applied rewrites 53.5%

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

   9. Applied rewrites 78.3%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in d around 0

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

   12. Applied rewrites 66.6%

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

   if 2.50000000000000022e-112 < (*.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)))) < 3.9999999999999997e250

   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 36.8

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

   5. Applied rewrites 36.8%

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

   7. Applied rewrites 99.6%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}, d\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \left(-0.125 \cdot \left(D \cdot D\right)\right) \cdot \frac{M \cdot M}{d}, d\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot -0.125\right) \cdot \left(\frac{M \cdot M}{\ell} \cdot \frac{h}{d}\right)\right)}{h}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (/
          (*
           (sqrt (/ h l))
           (* (* (* D_m D_m) -0.125) (* (/ (* M M) l) (/ h d))))
          h)))
   (if (<= t_0 -5e-138)
     t_1
     (if (<= t_0 4e+250)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (if (<= t_0 INFINITY) (fabs (/ d (sqrt (* l h)))) t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (sqrt((h / l)) * (((D_m * D_m) * -0.125) * (((M * M) / l) * (h / d)))) / h;
	double tmp;
	if (t_0 <= -5e-138) {
		tmp = t_1;
	} else if (t_0 <= 4e+250) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = (Math.sqrt((h / l)) * (((D_m * D_m) * -0.125) * (((M * M) / l) * (h / d)))) / h;
	double tmp;
	if (t_0 <= -5e-138) {
		tmp = t_1;
	} else if (t_0 <= 4e+250) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = (math.sqrt((h / l)) * (((D_m * D_m) * -0.125) * (((M * M) / l) * (h / d)))) / h
	tmp = 0
	if t_0 <= -5e-138:
		tmp = t_1
	elif t_0 <= 4e+250:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	elif t_0 <= math.inf:
		tmp = math.fabs((d / math.sqrt((l * h))))
	else:
		tmp = t_1
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(Float64(sqrt(Float64(h / l)) * Float64(Float64(Float64(D_m * D_m) * -0.125) * Float64(Float64(Float64(M * M) / l) * Float64(h / d)))) / h)
	tmp = 0.0
	if (t_0 <= -5e-138)
		tmp = t_1;
	elseif (t_0 <= 4e+250)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	elseif (t_0 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = (sqrt((h / l)) * (((D_m * D_m) * -0.125) * (((M * M) / l) * (h / d)))) / h;
	tmp = 0.0;
	if (t_0 <= -5e-138)
		tmp = t_1;
	elseif (t_0 <= 4e+250)
		tmp = sqrt((d / l)) / sqrt((h / d));
	elseif (t_0 <= Inf)
		tmp = abs((d / sqrt((l * h))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * -0.125), $MachinePrecision] * N[(N[(N[(M * M), $MachinePrecision] / l), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-138], t$95$1, If[LessEqual[t$95$0, 4e+250], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -4.99999999999999989e-138 or +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 62.1%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.9%

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

   7. Applied rewrites 50.9%

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

   9. Applied rewrites 78.2%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]

   10. Taylor expanded in d around 0

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

   12. Applied rewrites 62.2%

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

   if -4.99999999999999989e-138 < (*.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)))) < 3.9999999999999997e250

   1. Initial program 88.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.4

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

   5. Applied rewrites 38.4%

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

   7. Applied rewrites 88.9%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M \cdot M}{\ell} \cdot \frac{h}{d}\right)\right)}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \left(\left(\left(D \cdot D\right) \cdot -0.125\right) \cdot \left(\frac{M \cdot M}{\ell} \cdot \frac{h}{d}\right)\right)}{h}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{D\_m}{d} \cdot M\right)}^{2} \cdot 0.25, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(D\_m \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 4e+250)
     (*
      (*
       (fma (* (/ h l) -0.5) (* (pow (* (/ D_m d) M) 2.0) 0.25) 1.0)
       (sqrt (/ d l)))
      (sqrt (/ d h)))
     (if (<= t_0 INFINITY)
       (fabs (/ d (sqrt (* l h))))
       (*
        (/ (fma (/ (* (pow (* D_m M) 2.0) -0.125) d) (/ h l) d) h)
        (sqrt (/ h l)))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= 4e+250) {
		tmp = (fma(((h / l) * -0.5), (pow(((D_m / d) * M), 2.0) * 0.25), 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs((d / sqrt((l * h))));
	} else {
		tmp = (fma(((pow((D_m * M), 2.0) * -0.125) / d), (h / l), d) / h) * sqrt((h / l));
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= 4e+250)
		tmp = Float64(Float64(fma(Float64(Float64(h / l) * -0.5), Float64((Float64(Float64(D_m / d) * M) ^ 2.0) * 0.25), 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	elseif (t_0 <= Inf)
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(fma(Float64(Float64((Float64(D_m * M) ^ 2.0) * -0.125) / d), Float64(h / l), d) / h) * sqrt(Float64(h / l)));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e+250], N[(N[(N[(N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[Power[N[(N[(D$95$m / d), $MachinePrecision] * M), $MachinePrecision], 2.0], $MachinePrecision] * 0.25), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[(N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision] * -0.125), $MachinePrecision] / d), $MachinePrecision] * N[(h / l), $MachinePrecision] + d), $MachinePrecision] / h), $MachinePrecision] * N[Sqrt[N[(h / l), $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)))) < 3.9999999999999997e250

   1. Initial program 88.4%

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

   4. Applied rewrites 61.4%

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

   6. Applied rewrites 88.4%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 47.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 50.3

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 97.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 15.6%

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

   7. Applied rewrites 18.4%

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

   9. Applied rewrites 62.1%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{h}{\ell} \cdot -0.5, {\left(\frac{D}{d} \cdot M\right)}^{2} \cdot 0.25, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{\left(D \cdot M\right)}^{2} \cdot -0.125}{d}, \frac{h}{\ell}, d\right)}{h} \cdot \sqrt{\frac{h}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -5e-138)
     (/ (* (- d) (sqrt (/ h l))) h)
     (if (<= t_0 4e+250)
       (/ (sqrt (/ d l)) (sqrt (/ h d)))
       (fabs (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-138) {
		tmp = (-d * sqrt((h / l))) / h;
	} else if (t_0 <= 4e+250) {
		tmp = sqrt((d / l)) / sqrt((h / d));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    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 * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-5d-138)) then
        tmp = (-d * sqrt((h / l))) / h
    else if (t_0 <= 4d+250) then
        tmp = sqrt((d / l)) / sqrt((h / d))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-138) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else if (t_0 <= 4e+250) {
		tmp = Math.sqrt((d / l)) / Math.sqrt((h / d));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -5e-138:
		tmp = (-d * math.sqrt((h / l))) / h
	elif t_0 <= 4e+250:
		tmp = math.sqrt((d / l)) / math.sqrt((h / d))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-138)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	elseif (t_0 <= 4e+250)
		tmp = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -5e-138)
		tmp = (-d * sqrt((h / l))) / h;
	elseif (t_0 <= 4e+250)
		tmp = sqrt((d / l)) / sqrt((h / d));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-138], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 4e+250], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $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)))) < -4.99999999999999989e-138

   1. Initial program 87.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 15.1%

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

   if -4.99999999999999989e-138 < (*.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)))) < 3.9999999999999997e250

   1. Initial program 88.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.4

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

   5. Applied rewrites 38.4%

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

   7. Applied rewrites 88.9%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 25.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -5e-138)
     (/ (* (- d) (sqrt (/ h l))) h)
     (if (<= t_0 4e+250)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (fabs (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-138) {
		tmp = (-d * sqrt((h / l))) / h;
	} else if (t_0 <= 4e+250) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    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 * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-5d-138)) then
        tmp = (-d * sqrt((h / l))) / h
    else if (t_0 <= 4d+250) then
        tmp = sqrt((d / l)) * sqrt((d / h))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -5e-138) {
		tmp = (-d * Math.sqrt((h / l))) / h;
	} else if (t_0 <= 4e+250) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -5e-138:
		tmp = (-d * math.sqrt((h / l))) / h
	elif t_0 <= 4e+250:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -5e-138)
		tmp = Float64(Float64(Float64(-d) * sqrt(Float64(h / l))) / h);
	elseif (t_0 <= 4e+250)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -5e-138)
		tmp = (-d * sqrt((h / l))) / h;
	elseif (t_0 <= 4e+250)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-138], N[(N[((-d) * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$0, 4e+250], 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)))) < -4.99999999999999989e-138

   1. Initial program 87.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 15.1%

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

   if -4.99999999999999989e-138 < (*.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)))) < 3.9999999999999997e250

   1. Initial program 88.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 38.4

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

   5. Applied rewrites 38.4%

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

   7. Applied rewrites 88.9%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 25.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-d\right) \cdot t\_0}{h}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -5e-138)
     (/ (* (- d) t_0) h)
     (if (<= t_1 4e+250) (/ (* t_0 d) h) (fabs (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((h / l));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-138) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 4e+250) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((h / l))
    t_1 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-5d-138)) then
        tmp = (-d * t_0) / h
    else if (t_1 <= 4d+250) then
        tmp = (t_0 * d) / h
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -5e-138) {
		tmp = (-d * t_0) / h;
	} else if (t_1 <= 4e+250) {
		tmp = (t_0 * d) / h;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((h / l))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -5e-138:
		tmp = (-d * t_0) / h
	elif t_1 <= 4e+250:
		tmp = (t_0 * d) / h
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(h / l))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -5e-138)
		tmp = Float64(Float64(Float64(-d) * t_0) / h);
	elseif (t_1 <= 4e+250)
		tmp = Float64(Float64(t_0 * d) / h);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((h / l));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -5e-138)
		tmp = (-d * t_0) / h;
	elseif (t_1 <= 4e+250)
		tmp = (t_0 * d) / h;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-138], N[(N[((-d) * t$95$0), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 4e+250], N[(N[(t$95$0 * d), $MachinePrecision] / h), $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)))) < -4.99999999999999989e-138

   1. Initial program 87.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 15.1%

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

   if -4.99999999999999989e-138 < (*.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)))) < 3.9999999999999997e250

   1. Initial program 88.9%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 82.5%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 25.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{\left(-d\right) \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-135}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -4e-135)
     (* (- d) (sqrt (pow (* l h) -1.0)))
     (if (<= t_0 4e+250)
       (/ (* (sqrt (/ h l)) d) h)
       (fabs (/ d (sqrt (* l h))))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -4e-135) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else if (t_0 <= 4e+250) {
		tmp = (sqrt((h / l)) * d) / h;
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    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 * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-4d-135)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else if (t_0 <= 4d+250) then
        tmp = (sqrt((h / l)) * d) / h
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -4e-135) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else if (t_0 <= 4e+250) {
		tmp = (Math.sqrt((h / l)) * d) / h;
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -4e-135:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	elif t_0 <= 4e+250:
		tmp = (math.sqrt((h / l)) * d) / h
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -4e-135)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	elseif (t_0 <= 4e+250)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * d) / h);
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -4e-135)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	elseif (t_0 <= 4e+250)
		tmp = (sqrt((h / l)) * d) / h;
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-135], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+250], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * d), $MachinePrecision] / h), $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)))) < -4.0000000000000002e-135

   1. Initial program 87.8%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 10.7

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

   5. Applied rewrites 10.7%

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

   if -4.0000000000000002e-135 < (*.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)))) < 3.9999999999999997e250

   1. Initial program 89.0%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in d around inf

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

   8. Applied rewrites 81.5%

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

   if 3.9999999999999997e250 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 25.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 28.8

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-135}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 4 \cdot 10^{+250}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\frac{t\_0}{d}\right)}^{-1}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{t\_0}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1
         (*
          (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
          (-
           1.0
           (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 5e-170)
     (pow (/ t_0 d) -1.0)
     (if (<= t_1 2e+104) (sqrt (* (/ (/ d l) h) d)) (fabs (/ d t_0))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = sqrt((l * h));
	double t_1 = (pow((d / h), pow(2.0, -1.0)) * pow((d / l), pow(2.0, -1.0))) * (1.0 - ((pow(2.0, -1.0) * pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 5e-170) {
		tmp = pow((t_0 / d), -1.0);
	} else if (t_1 <= 2e+104) {
		tmp = sqrt((((d / l) / h) * d));
	} else {
		tmp = fabs((d / t_0));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * h))
    t_1 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_1 <= 5d-170) then
        tmp = (t_0 / d) ** (-1.0d0)
    else if (t_1 <= 2d+104) then
        tmp = sqrt((((d / l) / h) * d))
    else
        tmp = abs((d / t_0))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = (Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 5e-170) {
		tmp = Math.pow((t_0 / d), -1.0);
	} else if (t_1 <= 2e+104) {
		tmp = Math.sqrt((((d / l) / h) * d));
	} else {
		tmp = Math.abs((d / t_0));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	t_0 = math.sqrt((l * h))
	t_1 = (math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= 5e-170:
		tmp = math.pow((t_0 / d), -1.0)
	elif t_1 <= 2e+104:
		tmp = math.sqrt((((d / l) / h) * d))
	else:
		tmp = math.fabs((d / t_0))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64((Float64(d / h) ^ (2.0 ^ -1.0)) * (Float64(d / l) ^ (2.0 ^ -1.0))) * Float64(1.0 - Float64(Float64((2.0 ^ -1.0) * (Float64(Float64(M * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 5e-170)
		tmp = Float64(t_0 / d) ^ -1.0;
	elseif (t_1 <= 2e+104)
		tmp = sqrt(Float64(Float64(Float64(d / l) / h) * d));
	else
		tmp = abs(Float64(d / t_0));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = sqrt((l * h));
	t_1 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= 5e-170)
		tmp = (t_0 / d) ^ -1.0;
	elseif (t_1 <= 2e+104)
		tmp = sqrt((((d / l) / h) * d));
	else
		tmp = abs((d / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-170], N[Power[N[(t$95$0 / d), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, 2e+104], N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision], N[Abs[N[(d / t$95$0), $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)))) < 5.0000000000000001e-170

   1. Initial program 82.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 16.3

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

   5. Applied rewrites 16.3%

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

   7. Applied rewrites 16.3%

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

   9. Applied rewrites 16.3%

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

   if 5.0000000000000001e-170 < (*.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)))) < 2e104

   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 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 96.6%

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

   if 2e104 < (*.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 36.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 61.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\frac{\sqrt{\ell \cdot h}}{d}\right)}^{-1}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (/ d (sqrt (* l h)))))
   (if (<= t_0 5e-170)
     t_1
     (if (<= t_0 2e+104) (sqrt (* (/ (/ d l) h) d)) (fabs t_1)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / sqrt((l * h));
	double tmp;
	if (t_0 <= 5e-170) {
		tmp = t_1;
	} else if (t_0 <= 2e+104) {
		tmp = sqrt((((d / l) / h) * d));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    t_1 = d / sqrt((l * h))
    if (t_0 <= 5d-170) then
        tmp = t_1
    else if (t_0 <= 2d+104) then
        tmp = sqrt((((d / l) / h) * d))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = d / Math.sqrt((l * h));
	double tmp;
	if (t_0 <= 5e-170) {
		tmp = t_1;
	} else if (t_0 <= 2e+104) {
		tmp = Math.sqrt((((d / l) / h) * d));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = d / math.sqrt((l * h))
	tmp = 0
	if t_0 <= 5e-170:
		tmp = t_1
	elif t_0 <= 2e+104:
		tmp = math.sqrt((((d / l) / h) * d))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (t_0 <= 5e-170)
		tmp = t_1;
	elseif (t_0 <= 2e+104)
		tmp = sqrt(Float64(Float64(Float64(d / l) / h) * d));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	t_0 = (((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = d / sqrt((l * h));
	tmp = 0.0;
	if (t_0 <= 5e-170)
		tmp = t_1;
	elseif (t_0 <= 2e+104)
		tmp = sqrt((((d / l) / h) * d));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-170], t$95$1, If[LessEqual[t$95$0, 2e+104], N[Sqrt[N[(N[(N[(d / l), $MachinePrecision] / h), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.0000000000000001e-170

   1. Initial program 82.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 16.3

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

   5. Applied rewrites 16.3%

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

   7. Applied rewrites 16.3%

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

   if 5.0000000000000001e-170 < (*.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)))) < 2e104

   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 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 96.6%

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

   if 2e104 < (*.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 36.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 61.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{-170}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 2 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{\frac{\frac{d}{\ell}}{h} \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 45.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -4 \cdot 10^{-135}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (pow 2.0 -1.0)) (pow (/ d l) (pow 2.0 -1.0)))
       (-
        1.0
        (* (* (pow 2.0 -1.0) (pow (/ (* M D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -4e-135)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (fabs (/ d (sqrt (* l h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-135) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = fabs((d / sqrt((l * h))));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (((((d / h) ** (2.0d0 ** (-1.0d0))) * ((d / l) ** (2.0d0 ** (-1.0d0)))) * (1.0d0 - (((2.0d0 ** (-1.0d0)) * (((m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-4d-135)) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = abs((d / sqrt((l * h))))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (((Math.pow((d / h), Math.pow(2.0, -1.0)) * Math.pow((d / l), Math.pow(2.0, -1.0))) * (1.0 - ((Math.pow(2.0, -1.0) * Math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-135) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = Math.abs((d / Math.sqrt((l * h))));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if ((math.pow((d / h), math.pow(2.0, -1.0)) * math.pow((d / l), math.pow(2.0, -1.0))) * (1.0 - ((math.pow(2.0, -1.0) * math.pow(((M * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -4e-135:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = math.fabs((d / math.sqrt((l * h))))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -4e-135)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = abs(Float64(d / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (2.0 ^ -1.0)) * ((d / l) ^ (2.0 ^ -1.0))) * (1.0 - (((2.0 ^ -1.0) * (((M * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -4e-135)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = abs((d / sqrt((l * h))));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[Power[2.0, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[Power[2.0, -1.0], $MachinePrecision] * N[Power[N[(N[(M * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-135], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $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)))) < -4.0000000000000002e-135

   1. Initial program 87.8%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 10.7

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

   5. Applied rewrites 10.7%

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

   if -4.0000000000000002e-135 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 56.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 62.4%

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

4. Final simplification 43.6%

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

Alternative 16: 44.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :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 D_m) (* 2.0 d)) 2.0)) (/ h l))))
        -5e-138)
     t_0
     (fabs t_0))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-138) {
		tmp = t_0;
	} else {
		tmp = fabs(t_0);
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    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 * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-5d-138)) then
        tmp = t_0
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-138) {
		tmp = t_0;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	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 * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -5e-138:
		tmp = t_0
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	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 * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -5e-138)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	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 * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -5e-138)
		tmp = t_0;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := 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 * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-138], 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)))) < -4.99999999999999989e-138

   1. Initial program 87.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 10.3

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

   5. Applied rewrites 10.3%

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

   7. Applied rewrites 10.3%

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

   if -4.99999999999999989e-138 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 56.2%

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

   3. Taylor expanded in d around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 33.5

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

   5. Applied rewrites 33.5%

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

   7. Applied rewrites 62.8%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left({2}^{-1}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left({2}^{-1}\right)}\right) \cdot \left(1 - \left({2}^{-1} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5 \cdot \frac{h}{\ell}, {\left(\frac{d}{M} \cdot \frac{2}{D\_m}\right)}^{-2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -9 \cdot 10^{-57}:\\ \;\;\;\;\frac{t\_0 \cdot \sqrt{-d}}{\sqrt{-h}}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+205}:\\ \;\;\;\;\frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h \cdot {\left(D\_m \cdot M\right)}^{2}}{\ell}, \frac{-0.125}{d}, d\right)}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \sqrt{d}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (let* ((t_0
         (*
          (fma (* -0.5 (/ h l)) (pow (* (/ d M) (/ 2.0 D_m)) -2.0) 1.0)
          (sqrt (/ d l)))))
   (if (<= d -9e-57)
     (/ (* t_0 (sqrt (- d))) (sqrt (- h)))
     (if (<= d 6e+205)
       (/
        (* (sqrt (/ h l)) (fma (/ (* h (pow (* D_m M) 2.0)) l) (/ -0.125 d) d))
        h)
       (/ (* t_0 (sqrt d)) (sqrt h))))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double t_0 = fma((-0.5 * (h / l)), pow(((d / M) * (2.0 / D_m)), -2.0), 1.0) * sqrt((d / l));
	double tmp;
	if (d <= -9e-57) {
		tmp = (t_0 * sqrt(-d)) / sqrt(-h);
	} else if (d <= 6e+205) {
		tmp = (sqrt((h / l)) * fma(((h * pow((D_m * M), 2.0)) / l), (-0.125 / d), d)) / h;
	} else {
		tmp = (t_0 * sqrt(d)) / sqrt(h);
	}
	return tmp;
}

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	t_0 = Float64(fma(Float64(-0.5 * Float64(h / l)), (Float64(Float64(d / M) * Float64(2.0 / D_m)) ^ -2.0), 1.0) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (d <= -9e-57)
		tmp = Float64(Float64(t_0 * sqrt(Float64(-d))) / sqrt(Float64(-h)));
	elseif (d <= 6e+205)
		tmp = Float64(Float64(sqrt(Float64(h / l)) * fma(Float64(Float64(h * (Float64(D_m * M) ^ 2.0)) / l), Float64(-0.125 / d), d)) / h);
	else
		tmp = Float64(Float64(t_0 * sqrt(d)) / sqrt(h));
	end
	return tmp
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(-0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(d / M), $MachinePrecision] * N[(2.0 / D$95$m), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9e-57], N[(N[(t$95$0 * N[Sqrt[(-d)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6e+205], N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(h * N[Power[N[(D$95$m * M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(-0.125 / d), $MachinePrecision] + d), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * N[Sqrt[d], $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.99999999999999945e-57

   1. Initial program 76.0%

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

   4. Applied rewrites 89.7%

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

   if -8.99999999999999945e-57 < d < 5.9999999999999999e205

   1. Initial program 64.6%

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

   3. Taylor expanded in h around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 55.6%

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

   9. Applied rewrites 82.2%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, \frac{-0.125 \cdot {\left(M \cdot D\right)}^{2}}{d}, d\right)}{h} \]
   10. Step-by-step derivation

   11. Applied rewrites 83.8%

       \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot \mathsf{fma}\left(\frac{h \cdot {\left(D \cdot M\right)}^{2}}{\ell}, \frac{-0.125}{d}, d\right)}{h} \]

   if 5.9999999999999999e205 < d

   1. Initial program 60.9%

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

   4. Applied rewrites 99.4%

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

Alternative 18: 44.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} D_m = \left|D\right| \\ [d, h, l, M, D_m] = \mathsf{sort}([d, h, l, M, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.08 \cdot 10^{-173}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \end{array} \]

FPCore

D_m = (fabs.f64 D)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M D_m)
 :precision binary64
 (if (<= d 1.08e-173)
   (* (- d) (sqrt (pow (* l h) -1.0)))
   (/ d (* (sqrt l) (sqrt h)))))

C

D_m = fabs(D);
assert(d < h && h < l && l < M && M < D_m);
double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 1.08e-173) {
		tmp = -d * sqrt(pow((l * h), -1.0));
	} else {
		tmp = d / (sqrt(l) * sqrt(h));
	}
	return tmp;
}

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    real(8) :: tmp
    if (d <= 1.08d-173) then
        tmp = -d * sqrt(((l * h) ** (-1.0d0)))
    else
        tmp = d / (sqrt(l) * sqrt(h))
    end if
    code = tmp
end function

Java

D_m = Math.abs(D);
assert d < h && h < l && l < M && M < D_m;
public static double code(double d, double h, double l, double M, double D_m) {
	double tmp;
	if (d <= 1.08e-173) {
		tmp = -d * Math.sqrt(Math.pow((l * h), -1.0));
	} else {
		tmp = d / (Math.sqrt(l) * Math.sqrt(h));
	}
	return tmp;
}

Python

D_m = math.fabs(D)
[d, h, l, M, D_m] = sort([d, h, l, M, D_m])
def code(d, h, l, M, D_m):
	tmp = 0
	if d <= 1.08e-173:
		tmp = -d * math.sqrt(math.pow((l * h), -1.0))
	else:
		tmp = d / (math.sqrt(l) * math.sqrt(h))
	return tmp

Julia

D_m = abs(D)
d, h, l, M, D_m = sort([d, h, l, M, D_m])
function code(d, h, l, M, D_m)
	tmp = 0.0
	if (d <= 1.08e-173)
		tmp = Float64(Float64(-d) * sqrt((Float64(l * h) ^ -1.0)));
	else
		tmp = Float64(d / Float64(sqrt(l) * sqrt(h)));
	end
	return tmp
end

MATLAB

D_m = abs(D);
d, h, l, M, D_m = num2cell(sort([d, h, l, M, D_m])){:}
function tmp_2 = code(d, h, l, M, D_m)
	tmp = 0.0;
	if (d <= 1.08e-173)
		tmp = -d * sqrt(((l * h) ^ -1.0));
	else
		tmp = d / (sqrt(l) * sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D$95$m_] := If[LessEqual[d, 1.08e-173], N[((-d) * N[Sqrt[N[Power[N[(l * h), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.07999999999999995e-173

   1. Initial program 66.1%

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

   3. Taylor expanded in l around -inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 38.5

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

   5. Applied rewrites 38.5%

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

   if 1.07999999999999995e-173 < d

   1. Initial program 70.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 46.1%

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

   9. Applied rewrites 54.8%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.08 \cdot 10^{-173}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{{\left(\ell \cdot h\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 26.2% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

D_m = abs(d)
NOTE: d, h, l, M, and D_m should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_m)
    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_m
    code = d / sqrt((l * h))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
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 25.0

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

5. Applied rewrites 25.0%

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

7. Applied rewrites 25.0%

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

Reproduce

herbie shell --seed 2024322 
(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)))))