Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.8% → 79.3%

Time: 24.3s

Alternatives: 34

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := M\_m \cdot \left(D \cdot 0.5\right)\\ t_1 := \sqrt{h \cdot \ell}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ t_3 := \frac{M\_m \cdot D}{d \cdot 2}\\ t_4 := 1 + \frac{\frac{t\_0}{d \cdot 2}}{\ell} \cdot \frac{t\_3}{\frac{-1}{h}}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{-201}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_2\right) \cdot t\_4\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(t\_3, h \cdot \frac{t\_0}{2 \cdot \left(d \cdot \ell\right)}, 1\right)}{t\_1}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125, D \cdot \left(D \cdot \left(h \cdot \frac{M\_m \cdot M\_m}{d}\right)\right), d \cdot \ell\right)}{\ell}}{t\_1}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+122}:\\ \;\;\;\;t\_4 \cdot \left(t\_2 \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot D\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)} \cdot \left(M\_m \cdot \frac{D}{d \cdot 2}\right), h, 1\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0 (* M_m (* D 0.5)))
        (t_1 (sqrt (* h l)))
        (t_2 (sqrt (/ d l)))
        (t_3 (/ (* M_m D) (* d 2.0)))
        (t_4 (+ 1.0 (* (/ (/ t_0 (* d 2.0)) l) (/ t_3 (/ -1.0 h))))))
   (if (<= d -1.5e-201)
     (* (* (/ (sqrt (- d)) (sqrt (- h))) t_2) t_4)
     (if (<= d -5e-310)
       (/ (* d (fma t_3 (* h (/ t_0 (* 2.0 (* d l)))) 1.0)) t_1)
       (if (<= d 3.8e-133)
         (/ (/ (fma -0.125 (* D (* D (* h (/ (* M_m M_m) d)))) (* d l)) l) t_1)
         (if (<= d 3.4e+122)
           (* t_4 (* t_2 (sqrt (/ d h))))
           (/
            (*
             d
             (fma
              (*
               (/ (* (* M_m D) -0.5) (* l (* d 2.0)))
               (* M_m (/ D (* d 2.0))))
              h
              1.0))
            t_1)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = M_m * (D * 0.5);
	double t_1 = sqrt((h * l));
	double t_2 = sqrt((d / l));
	double t_3 = (M_m * D) / (d * 2.0);
	double t_4 = 1.0 + (((t_0 / (d * 2.0)) / l) * (t_3 / (-1.0 / h)));
	double tmp;
	if (d <= -1.5e-201) {
		tmp = ((sqrt(-d) / sqrt(-h)) * t_2) * t_4;
	} else if (d <= -5e-310) {
		tmp = (d * fma(t_3, (h * (t_0 / (2.0 * (d * l)))), 1.0)) / t_1;
	} else if (d <= 3.8e-133) {
		tmp = (fma(-0.125, (D * (D * (h * ((M_m * M_m) / d)))), (d * l)) / l) / t_1;
	} else if (d <= 3.4e+122) {
		tmp = t_4 * (t_2 * sqrt((d / h)));
	} else {
		tmp = (d * fma(((((M_m * D) * -0.5) / (l * (d * 2.0))) * (M_m * (D / (d * 2.0)))), h, 1.0)) / t_1;
	}
	return tmp;
}

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(M_m * Float64(D * 0.5))
	t_1 = sqrt(Float64(h * l))
	t_2 = sqrt(Float64(d / l))
	t_3 = Float64(Float64(M_m * D) / Float64(d * 2.0))
	t_4 = Float64(1.0 + Float64(Float64(Float64(t_0 / Float64(d * 2.0)) / l) * Float64(t_3 / Float64(-1.0 / h))))
	tmp = 0.0
	if (d <= -1.5e-201)
		tmp = Float64(Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_2) * t_4);
	elseif (d <= -5e-310)
		tmp = Float64(Float64(d * fma(t_3, Float64(h * Float64(t_0 / Float64(2.0 * Float64(d * l)))), 1.0)) / t_1);
	elseif (d <= 3.8e-133)
		tmp = Float64(Float64(fma(-0.125, Float64(D * Float64(D * Float64(h * Float64(Float64(M_m * M_m) / d)))), Float64(d * l)) / l) / t_1);
	elseif (d <= 3.4e+122)
		tmp = Float64(t_4 * Float64(t_2 * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(d * fma(Float64(Float64(Float64(Float64(M_m * D) * -0.5) / Float64(l * Float64(d * 2.0))) * Float64(M_m * Float64(D / Float64(d * 2.0)))), h, 1.0)) / t_1);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(M$95$m * N[(D * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(N[(N[(t$95$0 / N[(d * 2.0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(t$95$3 / N[(-1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.5e-201], N[(N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[d, -5e-310], N[(N[(d * N[(t$95$3 * N[(h * N[(t$95$0 / N[(2.0 * N[(d * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[d, 3.8e-133], N[(N[(N[(-0.125 * N[(D * N[(D * N[(h * N[(N[(M$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(d * l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[d, 3.4e+122], N[(t$95$4 * N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[(N[(N[(N[(M$95$m * D), $MachinePrecision] * -0.5), $MachinePrecision] / N[(l * N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(M$95$m * N[(D / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.50000000000000001e-201

   1. Initial program 78.7%

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

      1. lift-*.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. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 81.0

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

   6. Applied rewrites 81.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 81.0

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-neg.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 89.8

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

   8. Applied rewrites 89.8%

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

   if -1.50000000000000001e-201 < d < -4.999999999999985e-310

   1. Initial program 21.3%

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

   4. Applied rewrites 0.0%

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

   6. Applied rewrites 0.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 0.3%

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

   10. Applied rewrites 61.3%

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

   if -4.999999999999985e-310 < d < 3.8000000000000003e-133

   1. Initial program 37.8%

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

   4. Applied rewrites 11.8%

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

   6. Applied rewrites 37.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 43.3%

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

   9. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. associate-/l* N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 61.7

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

   11. Applied rewrites 61.7%

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

   if 3.8000000000000003e-133 < d < 3.4e122

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      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. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 87.3%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 87.3

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

   6. Applied rewrites 87.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 87.3

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

      3. lift-pow.f64 N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 87.3

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

   8. Applied rewrites 87.3%

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

   if 3.4e122 < d

   1. Initial program 73.7%

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

   4. Applied rewrites 29.5%

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

   6. Applied rewrites 60.1%

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

   8. Applied rewrites 95.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{\left(d \cdot 2\right) \cdot \ell} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right), h, 1\right)} \cdot d}{\sqrt{h \cdot \ell}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{-201}:\\ \;\;\;\;\left(\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 + \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{-1}{h}}\right)\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{M \cdot D}{d \cdot 2}, h \cdot \frac{M \cdot \left(D \cdot 0.5\right)}{2 \cdot \left(d \cdot \ell\right)}, 1\right)}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-0.125, D \cdot \left(D \cdot \left(h \cdot \frac{M \cdot M}{d}\right)\right), d \cdot \ell\right)}{\ell}}{\sqrt{h \cdot \ell}}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+122}:\\ \;\;\;\;\left(1 + \frac{\frac{M \cdot \left(D \cdot 0.5\right)}{d \cdot 2}}{\ell} \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\frac{-1}{h}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(\frac{\left(M \cdot D\right) \cdot -0.5}{\ell \cdot \left(d \cdot 2\right)} \cdot \left(M \cdot \frac{D}{d \cdot 2}\right), h, 1\right)}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ [d, h, l, M_m, D] = \mathsf{sort}([d, h, l, M_m, D])\\ \\ \begin{array}{l} t_0 := \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M\_m \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(-0.125 \cdot \frac{M\_m \cdot \left(M\_m \cdot \left(D \cdot D\right)\right)}{-d}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M\_m \cdot M\_m\right)\right)\right)}{d \cdot \ell}}{\sqrt{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D)
 :precision binary64
 (let* ((t_0
         (*
          (+
           1.0
           (* (/ h l) (* (pow (/ (* M_m D) (* d 2.0)) 2.0) (/ -1.0 2.0))))
          (* (pow (/ d l) (/ 1.0 2.0)) (pow (/ d h) (/ 1.0 2.0))))))
   (if (<= t_0 -1e+247)
     (*
      (sqrt (/ h (* l (* l l))))
      (* -0.125 (/ (* M_m (* M_m (* D D))) (- d))))
     (if (<= t_0 INFINITY)
       (* (sqrt (/ d l)) (sqrt (/ d h)))
       (/
        (/ (* (* D D) (* -0.125 (* h (* M_m M_m)))) (* d l))
        (sqrt (* h l)))))))

C

M_m = fabs(M);
assert(d < h && h < l && l < M_m && M_m < D);
double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (1.0 + ((h / l) * (pow(((M_m * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (pow((d / l), (1.0 / 2.0)) * pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= -1e+247) {
		tmp = sqrt((h / (l * (l * l)))) * (-0.125 * ((M_m * (M_m * (D * D))) / -d));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else {
		tmp = (((D * D) * (-0.125 * (h * (M_m * M_m)))) / (d * l)) / sqrt((h * l));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
assert d < h && h < l && l < M_m && M_m < D;
public static double code(double d, double h, double l, double M_m, double D) {
	double t_0 = (1.0 + ((h / l) * (Math.pow(((M_m * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (Math.pow((d / l), (1.0 / 2.0)) * Math.pow((d / h), (1.0 / 2.0)));
	double tmp;
	if (t_0 <= -1e+247) {
		tmp = Math.sqrt((h / (l * (l * l)))) * (-0.125 * ((M_m * (M_m * (D * D))) / -d));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else {
		tmp = (((D * D) * (-0.125 * (h * (M_m * M_m)))) / (d * l)) / Math.sqrt((h * l));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
[d, h, l, M_m, D] = sort([d, h, l, M_m, D])
def code(d, h, l, M_m, D):
	t_0 = (1.0 + ((h / l) * (math.pow(((M_m * D) / (d * 2.0)), 2.0) * (-1.0 / 2.0)))) * (math.pow((d / l), (1.0 / 2.0)) * math.pow((d / h), (1.0 / 2.0)))
	tmp = 0
	if t_0 <= -1e+247:
		tmp = math.sqrt((h / (l * (l * l)))) * (-0.125 * ((M_m * (M_m * (D * D))) / -d))
	elif t_0 <= math.inf:
		tmp = math.sqrt((d / l)) * math.sqrt((d / h))
	else:
		tmp = (((D * D) * (-0.125 * (h * (M_m * M_m)))) / (d * l)) / math.sqrt((h * l))
	return tmp

Julia

M_m = abs(M)
d, h, l, M_m, D = sort([d, h, l, M_m, D])
function code(d, h, l, M_m, D)
	t_0 = Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M_m * D) / Float64(d * 2.0)) ^ 2.0) * Float64(-1.0 / 2.0)))) * Float64((Float64(d / l) ^ Float64(1.0 / 2.0)) * (Float64(d / h) ^ Float64(1.0 / 2.0))))
	tmp = 0.0
	if (t_0 <= -1e+247)
		tmp = Float64(sqrt(Float64(h / Float64(l * Float64(l * l)))) * Float64(-0.125 * Float64(Float64(M_m * Float64(M_m * Float64(D * D))) / Float64(-d))));
	elseif (t_0 <= Inf)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(Float64(Float64(D * D) * Float64(-0.125 * Float64(h * Float64(M_m * M_m)))) / Float64(d * l)) / sqrt(Float64(h * l)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
d, h, l, M_m, D = num2cell(sort([d, h, l, M_m, D])){:}
function tmp_2 = code(d, h, l, M_m, D)
	t_0 = (1.0 + ((h / l) * ((((M_m * D) / (d * 2.0)) ^ 2.0) * (-1.0 / 2.0)))) * (((d / l) ^ (1.0 / 2.0)) * ((d / h) ^ (1.0 / 2.0)));
	tmp = 0.0;
	if (t_0 <= -1e+247)
		tmp = sqrt((h / (l * (l * l)))) * (-0.125 * ((M_m * (M_m * (D * D))) / -d));
	elseif (t_0 <= Inf)
		tmp = sqrt((d / l)) * sqrt((d / h));
	else
		tmp = (((D * D) * (-0.125 * (h * (M_m * M_m)))) / (d * l)) / sqrt((h * l));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
NOTE: d, h, l, M_m, and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D_] := Block[{t$95$0 = N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M$95$m * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+247], N[(N[Sqrt[N[(h / N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(M$95$m * N[(M$95$m * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(D * D), $MachinePrecision] * N[(-0.125 * N[(h * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d * l), $MachinePrecision]), $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)))) < -9.99999999999999952e246

   1. Initial program 84.2%

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in h around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

      17. mul-1-neg N/A

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

   7. Applied rewrites 37.6%

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

   if -9.99999999999999952e246 < (*.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 80.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 39.4

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 69.3%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 3.2%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in D around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 19.5

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

   9. Applied rewrites 19.5%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq -1 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{\frac{h}{\ell \cdot \left(\ell \cdot \ell\right)}} \cdot \left(-0.125 \cdot \frac{M \cdot \left(M \cdot \left(D \cdot D\right)\right)}{-d}\right)\\ \mathbf{elif}\;\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{-1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \leq \infty:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(D \cdot D\right) \cdot \left(-0.125 \cdot \left(h \cdot \left(M \cdot M\right)\right)\right)}{d \cdot \ell}}{\sqrt{h \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024229 
(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)))))