Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.7% → 76.5%

Time: 34.1s

Alternatives: 21

Speedup: 1.0×

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: 65.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ t_1 := \frac{h}{\ell} \cdot -0.5\\ t_2 := \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_1, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;\frac{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, t_1, 1\right)}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (+ 1.0 (/ (* h (* -0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l)))))
        (t_1 (* (/ h l) -0.5))
        (t_2
         (*
          (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_1 1.0)
          (/ (- d) (sqrt (* l h))))))
   (if (<= d -2.7e+131)
     t_2
     (if (<= d -1.3e-116)
       t_0
       (if (<= d -1.55e-276)
         t_2
         (if (<= d 1.9e-51)
           (/
            (* d (/ (fma (pow (* D (/ (* M 0.5) d)) 2.0) t_1 1.0) (sqrt h)))
            (sqrt l))
           (if (<= d 5e+64)
             t_0
             (*
              (fma (/ h l) (* -0.5 (pow (* D (/ M (* d 2.0))) 2.0)) 1.0)
              (/ d (* (sqrt h) (sqrt l)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)));
	double t_1 = (h / l) * -0.5;
	double t_2 = fma(pow(((M * 0.5) / (d / D)), 2.0), t_1, 1.0) * (-d / sqrt((l * h)));
	double tmp;
	if (d <= -2.7e+131) {
		tmp = t_2;
	} else if (d <= -1.3e-116) {
		tmp = t_0;
	} else if (d <= -1.55e-276) {
		tmp = t_2;
	} else if (d <= 1.9e-51) {
		tmp = (d * (fma(pow((D * ((M * 0.5) / d)), 2.0), t_1, 1.0) / sqrt(h))) / sqrt(l);
	} else if (d <= 5e+64) {
		tmp = t_0;
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l))))
	t_1 = Float64(Float64(h / l) * -0.5)
	t_2 = Float64(fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_1, 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (d <= -2.7e+131)
		tmp = t_2;
	elseif (d <= -1.3e-116)
		tmp = t_0;
	elseif (d <= -1.55e-276)
		tmp = t_2;
	elseif (d <= 1.9e-51)
		tmp = Float64(Float64(d * Float64(fma((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0), t_1, 1.0) / sqrt(h))) / sqrt(l));
	elseif (d <= 5e+64)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.7e+131], t$95$2, If[LessEqual[d, -1.3e-116], t$95$0, If[LessEqual[d, -1.55e-276], t$95$2, If[LessEqual[d, 1.9e-51], N[(N[(d * N[(N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+64], t$95$0, N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.70000000000000004e131 or -1.3e-116 < d < -1.54999999999999995e-276

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

      1. frac-times 53.7%

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

      2. associate-/r* 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. expm1-log1p-u 27.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 17.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 18.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 40.6%

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

      3. associate-*r* 40.6%

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

      4. *-commutative 40.6%

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

      5. *-commutative 40.6%

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

      6. +-commutative 40.6%

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

      7. fma-def 40.6%

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

      8. *-commutative 40.6%

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

      9. associate-/r/ 42.1%

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

      10. *-commutative 42.1%

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

   10. Simplified 42.1%

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

   11. Taylor expanded in d around -inf 72.9%

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

       1. mul-1-neg 72.9%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 72.9%

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

   if -2.70000000000000004e131 < d < -1.3e-116 or 1.90000000000000001e-51 < d < 5e64

   1. Initial program 82.1%

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

   3. Simplified 82.1%

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

      1. associate-*l/ 92.3%

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

   6. Applied egg-rr 92.3%

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

   if -1.54999999999999995e-276 < d < 1.90000000000000001e-51

   1. Initial program 48.0%

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

   3. Simplified 46.0%

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

   6. Applied egg-rr 10.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 63.5%

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

      1. associate-*l/ 73.9%

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

      2. associate-/r* 73.9%

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

      3. div-inv 73.9%

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

      4. metadata-eval 73.9%

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

   10. Applied egg-rr 73.9%

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

   if 5e64 < d

   1. Initial program 77.0%

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

   3. Simplified 79.0%

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

   6. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 91.7%

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

      3. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;\frac{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ t_1 := \frac{h}{\ell} \cdot -0.5\\ t_2 := \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_1, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-66}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, t_1, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (+ 1.0 (/ (* h (* -0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l)))))
        (t_1 (* (/ h l) -0.5))
        (t_2
         (*
          (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_1 1.0)
          (/ (- d) (sqrt (* l h))))))
   (if (<= d -2.7e+132)
     t_2
     (if (<= d -1.22e-119)
       t_0
       (if (<= d -1.55e-276)
         t_2
         (if (<= d 7e-66)
           (/
            (* d (fma (pow (* D (/ (* M 0.5) d)) 2.0) t_1 1.0))
            (pow (* l h) 0.5))
           (if (<= d 3e+65)
             t_0
             (*
              (fma (/ h l) (* -0.5 (pow (* D (/ M (* d 2.0))) 2.0)) 1.0)
              (/ d (* (sqrt h) (sqrt l)))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)));
	double t_1 = (h / l) * -0.5;
	double t_2 = fma(pow(((M * 0.5) / (d / D)), 2.0), t_1, 1.0) * (-d / sqrt((l * h)));
	double tmp;
	if (d <= -2.7e+132) {
		tmp = t_2;
	} else if (d <= -1.22e-119) {
		tmp = t_0;
	} else if (d <= -1.55e-276) {
		tmp = t_2;
	} else if (d <= 7e-66) {
		tmp = (d * fma(pow((D * ((M * 0.5) / d)), 2.0), t_1, 1.0)) / pow((l * h), 0.5);
	} else if (d <= 3e+65) {
		tmp = t_0;
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l))))
	t_1 = Float64(Float64(h / l) * -0.5)
	t_2 = Float64(fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_1, 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (d <= -2.7e+132)
		tmp = t_2;
	elseif (d <= -1.22e-119)
		tmp = t_0;
	elseif (d <= -1.55e-276)
		tmp = t_2;
	elseif (d <= 7e-66)
		tmp = Float64(Float64(d * fma((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0), t_1, 1.0)) / (Float64(l * h) ^ 0.5));
	elseif (d <= 3e+65)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.7e+132], t$95$2, If[LessEqual[d, -1.22e-119], t$95$0, If[LessEqual[d, -1.55e-276], t$95$2, If[LessEqual[d, 7e-66], N[(N[(d * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3e+65], t$95$0, N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.7e132 or -1.22e-119 < d < -1.54999999999999995e-276

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

      1. frac-times 53.7%

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

      2. associate-/r* 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. expm1-log1p-u 27.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 17.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 18.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 40.6%

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

      3. associate-*r* 40.6%

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

      4. *-commutative 40.6%

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

      5. *-commutative 40.6%

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

      6. +-commutative 40.6%

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

      7. fma-def 40.6%

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

      8. *-commutative 40.6%

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

      9. associate-/r/ 42.1%

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

      10. *-commutative 42.1%

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

   10. Simplified 42.1%

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

   11. Taylor expanded in d around -inf 72.9%

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

       1. mul-1-neg 72.9%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 72.9%

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

   if -2.7e132 < d < -1.22e-119 or 7.0000000000000001e-66 < d < 3.0000000000000002e65

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

      1. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

   if -1.54999999999999995e-276 < d < 7.0000000000000001e-66

   1. Initial program 46.9%

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

   3. Simplified 44.8%

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

   6. Applied egg-rr 10.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 62.7%

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

      1. frac-times 73.2%

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

      2. associate-/r* 73.2%

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

      3. div-inv 73.2%

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

      4. metadata-eval 73.2%

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

      5. pow1/2 73.2%

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

      6. pow1/2 73.2%

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

      7. pow-prod-down 68.2%

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

   10. Applied egg-rr 68.2%

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

   if 3.0000000000000002e65 < d

   1. Initial program 77.0%

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

   3. Simplified 79.0%

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

   6. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 91.7%

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

      3. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{-119}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-66}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}}\\ t_1 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{-79}:\\ \;\;\;\;\left(t_0 \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_1, 1\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, t_1, 1\right)}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d h))) (t_1 (* (/ h l) -0.5)))
   (if (<= d -1.32e-79)
     (*
      (* t_0 (/ (sqrt (- d)) (sqrt (- l))))
      (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))
     (if (<= d -1.55e-276)
       (*
        (* d (- (sqrt (/ (/ 1.0 l) h))))
        (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_1 1.0))
       (if (<= d 9.2e-58)
         (/
          (* d (/ (fma (pow (* D (/ (* M 0.5) d)) 2.0) t_1 1.0) (sqrt h)))
          (sqrt l))
         (if (<= d 9.5e+62)
           (*
            t_0
            (*
             (sqrt (/ d l))
             (+ 1.0 (/ (* h (* -0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l))))
           (*
            (fma (/ h l) (* -0.5 (pow (* D (/ M (* d 2.0))) 2.0)) 1.0)
            (/ d (* (sqrt h) (sqrt l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h));
	double t_1 = (h / l) * -0.5;
	double tmp;
	if (d <= -1.32e-79) {
		tmp = (t_0 * (sqrt(-d) / sqrt(-l))) * (1.0 - (0.5 * ((h / l) * pow(((M / 2.0) * (D / d)), 2.0))));
	} else if (d <= -1.55e-276) {
		tmp = (d * -sqrt(((1.0 / l) / h))) * fma(pow(((M * 0.5) / (d / D)), 2.0), t_1, 1.0);
	} else if (d <= 9.2e-58) {
		tmp = (d * (fma(pow((D * ((M * 0.5) / d)), 2.0), t_1, 1.0) / sqrt(h))) / sqrt(l);
	} else if (d <= 9.5e+62) {
		tmp = t_0 * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)));
	} else {
		tmp = fma((h / l), (-0.5 * pow((D * (M / (d * 2.0))), 2.0)), 1.0) * (d / (sqrt(h) * sqrt(l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / h))
	t_1 = Float64(Float64(h / l) * -0.5)
	tmp = 0.0
	if (d <= -1.32e-79)
		tmp = Float64(Float64(t_0 * Float64(sqrt(Float64(-d)) / sqrt(Float64(-l)))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0)))));
	elseif (d <= -1.55e-276)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h)))) * fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_1, 1.0));
	elseif (d <= 9.2e-58)
		tmp = Float64(Float64(d * Float64(fma((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0), t_1, 1.0) / sqrt(h))) / sqrt(l));
	elseif (d <= 9.5e+62)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l))));
	else
		tmp = Float64(fma(Float64(h / l), Float64(-0.5 * (Float64(D * Float64(M / Float64(d * 2.0))) ^ 2.0)), 1.0) * Float64(d / Float64(sqrt(h) * sqrt(l))));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[d, -1.32e-79], N[(N[(t$95$0 * N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.55e-276], N[(N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-58], N[(N[(d * N[(N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.5e+62], N[(t$95$0 * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(D * N[(M / N[(d * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.32e-79

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

      1. frac-2neg 77.7%

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

      2. sqrt-div 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -1.32e-79 < d < -1.54999999999999995e-276

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

      1. frac-times 49.8%

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

      2. associate-/r* 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. expm1-log1p-u 17.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 4.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 10.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 37.3%

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

      3. associate-*r* 37.3%

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

      4. *-commutative 37.3%

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

      5. *-commutative 37.3%

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

      6. +-commutative 37.3%

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

      7. fma-def 37.3%

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

      8. *-commutative 37.3%

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

      9. associate-/r/ 39.2%

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

      10. *-commutative 39.2%

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

   10. Simplified 39.2%

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

   11. Taylor expanded in d around -inf 68.6%

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

       1. mul-1-neg 68.6%

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

       2. distribute-rgt-neg-in 68.6%

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

       3. *-commutative 68.6%

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

       4. associate-/r* 68.6%

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

   13. Simplified 68.6%

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

   if -1.54999999999999995e-276 < d < 9.1999999999999995e-58

   1. Initial program 48.0%

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

   3. Simplified 46.0%

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

   6. Applied egg-rr 10.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 63.5%

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

      1. associate-*l/ 73.9%

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

      2. associate-/r* 73.9%

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

      3. div-inv 73.9%

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

      4. metadata-eval 73.9%

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

   10. Applied egg-rr 73.9%

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

   if 9.1999999999999995e-58 < d < 9.5000000000000003e62

   1. Initial program 73.8%

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

   3. Simplified 73.6%

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

      1. associate-*l/ 99.1%

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

   6. Applied egg-rr 99.1%

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

   if 9.5000000000000003e62 < d

   1. Initial program 77.0%

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

   3. Simplified 79.0%

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

   6. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 66.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 91.7%

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

      3. *-commutative 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{h}} \cdot \frac{\sqrt{-d}}{\sqrt{-\ell}}\right) \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}}{\sqrt{\ell}}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{h}{\ell}, -0.5 \cdot {\left(D \cdot \frac{M}{d \cdot 2}\right)}^{2}, 1\right) \cdot \frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.85e-302)
   (*
    (/ (sqrt (- d)) (sqrt (- h)))
    (*
     (sqrt (/ d l))
     (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5)))))
   (/
    (*
     d
     (/ (fma (pow (* D (/ (* M 0.5) d)) 2.0) (* (/ h l) -0.5) 1.0) (sqrt h)))
    (sqrt l))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.85e-302) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5))));
	} else {
		tmp = (d * (fma(pow((D * ((M * 0.5) / d)), 2.0), ((h / l) * -0.5), 1.0) / sqrt(h))) / sqrt(l);
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.85e-302)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))));
	else
		tmp = Float64(Float64(d * Float64(fma((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) / sqrt(h))) / sqrt(l));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.85e-302], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.85e-302

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

      1. frac-2neg 66.4%

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

      2. sqrt-div 79.8%

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

   6. Applied egg-rr 79.8%

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

   if -1.85e-302 < l

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   6. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 77.4%

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

      1. associate-*l/ 80.6%

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

      2. associate-/r* 80.6%

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

      3. div-inv 80.6%

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

      4. metadata-eval 80.6%

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

   10. Applied egg-rr 80.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ t_1 := \frac{h}{\ell} \cdot -0.5\\ t_2 := {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\\ t_3 := \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_1, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+130}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-66}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left(t_2, t_1, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + t_2 \cdot t_1}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d h))
          (*
           (sqrt (/ d l))
           (+ 1.0 (/ (* h (* -0.5 (pow (* (/ D d) (* M 0.5)) 2.0))) l)))))
        (t_1 (* (/ h l) -0.5))
        (t_2 (pow (* D (/ (* M 0.5) d)) 2.0))
        (t_3
         (*
          (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_1 1.0)
          (/ (- d) (sqrt (* l h))))))
   (if (<= d -2.3e+130)
     t_3
     (if (<= d -2.9e-118)
       t_0
       (if (<= d -1.55e-276)
         t_3
         (if (<= d 1.75e-66)
           (/ (* d (fma t_2 t_1 1.0)) (pow (* l h) 0.5))
           (if (<= d 7.2e+62)
             t_0
             (* (/ d (sqrt l)) (/ (+ 1.0 (* t_2 t_1)) (sqrt h))))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / h)) * (sqrt((d / l)) * (1.0 + ((h * (-0.5 * pow(((D / d) * (M * 0.5)), 2.0))) / l)));
	double t_1 = (h / l) * -0.5;
	double t_2 = pow((D * ((M * 0.5) / d)), 2.0);
	double t_3 = fma(pow(((M * 0.5) / (d / D)), 2.0), t_1, 1.0) * (-d / sqrt((l * h)));
	double tmp;
	if (d <= -2.3e+130) {
		tmp = t_3;
	} else if (d <= -2.9e-118) {
		tmp = t_0;
	} else if (d <= -1.55e-276) {
		tmp = t_3;
	} else if (d <= 1.75e-66) {
		tmp = (d * fma(t_2, t_1, 1.0)) / pow((l * h), 0.5);
	} else if (d <= 7.2e+62) {
		tmp = t_0;
	} else {
		tmp = (d / sqrt(l)) * ((1.0 + (t_2 * t_1)) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h * Float64(-0.5 * (Float64(Float64(D / d) * Float64(M * 0.5)) ^ 2.0))) / l))))
	t_1 = Float64(Float64(h / l) * -0.5)
	t_2 = Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0
	t_3 = Float64(fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_1, 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))))
	tmp = 0.0
	if (d <= -2.3e+130)
		tmp = t_3;
	elseif (d <= -2.9e-118)
		tmp = t_0;
	elseif (d <= -1.55e-276)
		tmp = t_3;
	elseif (d <= 1.75e-66)
		tmp = Float64(Float64(d * fma(t_2, t_1, 1.0)) / (Float64(l * h) ^ 0.5));
	elseif (d <= 7.2e+62)
		tmp = t_0;
	else
		tmp = Float64(Float64(d / sqrt(l)) * Float64(Float64(1.0 + Float64(t_2 * t_1)) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(-0.5 * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M * 0.5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.3e+130], t$95$3, If[LessEqual[d, -2.9e-118], t$95$0, If[LessEqual[d, -1.55e-276], t$95$3, If[LessEqual[d, 1.75e-66], N[(N[(d * N[(t$95$2 * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e+62], t$95$0, N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.30000000000000021e130 or -2.8999999999999998e-118 < d < -1.54999999999999995e-276

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

      1. frac-times 53.7%

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

      2. associate-/r* 53.7%

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

   6. Applied egg-rr 53.7%

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

      1. expm1-log1p-u 27.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 17.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 18.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 40.6%

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

      3. associate-*r* 40.6%

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

      4. *-commutative 40.6%

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

      5. *-commutative 40.6%

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

      6. +-commutative 40.6%

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

      7. fma-def 40.6%

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

      8. *-commutative 40.6%

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

      9. associate-/r/ 42.1%

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

      10. *-commutative 42.1%

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

   10. Simplified 42.1%

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

   11. Taylor expanded in d around -inf 72.9%

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

       1. mul-1-neg 72.9%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 72.9%

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

   if -2.30000000000000021e130 < d < -2.8999999999999998e-118 or 1.75e-66 < d < 7.2e62

   1. Initial program 82.3%

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

   3. Simplified 82.3%

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

      1. associate-*l/ 92.4%

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

   6. Applied egg-rr 92.4%

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

   if -1.54999999999999995e-276 < d < 1.75e-66

   1. Initial program 46.9%

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

   3. Simplified 44.8%

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

   6. Applied egg-rr 10.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 62.7%

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

      1. frac-times 73.2%

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

      2. associate-/r* 73.2%

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

      3. div-inv 73.2%

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

      4. metadata-eval 73.2%

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

      5. pow1/2 73.2%

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

      6. pow1/2 73.2%

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

      7. pow-prod-down 68.2%

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

   10. Applied egg-rr 68.2%

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

   if 7.2e62 < d

   1. Initial program 77.0%

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

   3. Simplified 79.0%

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

   6. Applied egg-rr 60.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 91.7%

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

      1. fma-udef 91.7%

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

      2. associate-/r* 91.7%

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

      3. div-inv 91.7%

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

      4. metadata-eval 91.7%

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

   10. Applied egg-rr 91.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-66}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+62}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h \cdot \left(-0.5 \cdot {\left(\frac{D}{d} \cdot \left(M \cdot 0.5\right)\right)}^{2}\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{+188}:\\ \;\;\;\;\left|\frac{d}{t_0}\right|\\ \mathbf{elif}\;d \leq -7.6 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-293}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\frac{t_0}{\mathsf{fma}\left({\left(\frac{0.5}{\frac{\frac{d}{D}}{M}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= d -4.4e+188)
     (fabs (/ d t_0))
     (if (<= d -7.6e-221)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (+ 1.0 (* (pow (* D (/ (* M 0.5) d)) 2.0) (* (/ h l) -0.5))))
       (if (<= d 2.4e-293)
         (* d (log (exp (pow (* l h) -0.5))))
         (if (<= d 1.7e+127)
           (/
            d
            (/ t_0 (fma (pow (/ 0.5 (/ (/ d D) M)) 2.0) (/ (* h -0.5) l) 1.0)))
           (* d (/ (pow l -0.5) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (d <= -4.4e+188) {
		tmp = fabs((d / t_0));
	} else if (d <= -7.6e-221) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= 2.4e-293) {
		tmp = d * log(exp(pow((l * h), -0.5)));
	} else if (d <= 1.7e+127) {
		tmp = d / (t_0 / fma(pow((0.5 / ((d / D) / M)), 2.0), ((h * -0.5) / l), 1.0));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (d <= -4.4e+188)
		tmp = abs(Float64(d / t_0));
	elseif (d <= -7.6e-221)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (d <= 2.4e-293)
		tmp = Float64(d * log(exp((Float64(l * h) ^ -0.5))));
	elseif (d <= 1.7e+127)
		tmp = Float64(d / Float64(t_0 / fma((Float64(0.5 / Float64(Float64(d / D) / M)) ^ 2.0), Float64(Float64(h * -0.5) / l), 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -4.4e+188], N[Abs[N[(d / t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -7.6e-221], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e-293], N[(d * N[Log[N[Exp[N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+127], N[(d / N[(t$95$0 / N[(N[Power[N[(0.5 / N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.39999999999999998e188

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in d around inf 9.5%

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

      1. *-commutative 9.5%

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

      2. associate-/r* 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in d around 0 9.5%

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

      1. unpow1/2 9.5%

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

      2. rem-exp-log 9.5%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 9.5%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 9.5%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 9.5%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 9.5%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 9.5%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 9.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 9.5%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-unprod 19.4%

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

       3. pow2 19.4%

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

   12. Applied egg-rr 19.4%

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

       1. unpow2 19.4%

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

       2. rem-sqrt-square 64.2%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       3. rem-exp-log 0.4%

          \[\leadsto \left|\color{blue}{e^{\log \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

       4. log-prod 0.0%

          \[\leadsto \left|e^{\color{blue}{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

       5. log-pow 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}}\right| \]

       6. metadata-eval 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5\right)} \cdot \log \left(h \cdot \ell\right)}\right| \]

       7. distribute-lft-neg-in 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}}\right| \]

       8. log-pow 0.0%

          \[\leadsto \left|e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)}\right| \]

       9. unpow1/2 0.0%

          \[\leadsto \left|e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)}\right| \]

       10. sub-neg 0.0%

           \[\leadsto \left|e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}}\right| \]

       11. log-div 0.4%

           \[\leadsto \left|e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}}\right| \]

       12. rem-exp-log 64.2%

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

   14. Simplified 64.2%

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

   if -4.39999999999999998e188 < d < -7.6000000000000002e-221

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

      1. frac-times 74.7%

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

      2. associate-/r* 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. expm1-log1p-u 35.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 20.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 26.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 62.5%

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

      3. associate-*r* 62.5%

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

      4. *-commutative 62.5%

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

      5. *-commutative 62.5%

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

      6. +-commutative 62.5%

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

      7. fma-def 62.5%

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

      8. *-commutative 62.5%

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

      9. associate-/r/ 63.7%

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

      10. *-commutative 63.7%

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

   10. Simplified 63.7%

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

       1. fma-udef 63.7%

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

       2. associate-/r/ 62.5%

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

       3. *-commutative 62.5%

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

   12. Applied egg-rr 62.5%

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

   if -7.6000000000000002e-221 < d < 2.3999999999999999e-293

   1. Initial program 29.5%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in d around inf 12.3%

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

      1. *-commutative 12.3%

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

      2. associate-/r* 12.3%

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

   7. Simplified 12.3%

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

   8. Taylor expanded in d around 0 12.3%

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

      1. unpow1/2 12.3%

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

      2. rem-exp-log 12.3%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 12.3%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 12.3%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 12.3%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 12.3%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 12.3%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 12.3%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 12.3%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-log-exp 40.7%

          \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   12. Applied egg-rr 40.7%

       \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   if 2.3999999999999999e-293 < d < 1.69999999999999989e127

   1. Initial program 62.0%

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

   3. Simplified 60.7%

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

   6. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 73.3%

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

      1. expm1-log1p-u 36.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell}} \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\right)\right)} \]

      2. expm1-udef 16.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell}} \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\right)} - 1} \]

   10. Applied egg-rr 14.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 32.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\right)\right)} \]

       2. expm1-log1p 70.6%

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

       3. associate-/l* 69.5%

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

       4. unpow1/2 69.5%

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

       5. *-commutative 69.5%

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

       6. *-commutative 69.5%

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

       7. associate-/r/ 69.1%

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

       8. *-commutative 69.1%

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

       9. associate-/l* 69.2%

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

       10. associate-*l/ 69.2%

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

   12. Simplified 69.2%

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

   if 1.69999999999999989e127 < d

   1. Initial program 74.7%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in d around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-/r* 62.6%

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

   7. Simplified 62.6%

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

      1. expm1-log1p-u 61.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 42.5%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 44.6%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 82.6%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 83.9%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 83.9%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+188}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -7.6 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-293}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell \cdot h}}{\mathsf{fma}\left({\left(\frac{0.5}{\frac{\frac{d}{D}}{M}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ t_1 := \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_0, 1\right)\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;t_1 \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot t_0}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5))
        (t_1 (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_0 1.0)))
   (if (<= d -3.8e+133)
     (* t_1 (/ (- d) (sqrt (* l h))))
     (if (<= d -1.75e-79)
       (*
        (*
         (sqrt (/ d l))
         (+ 1.0 (* (/ h l) (* (pow (* (/ M 2.0) (/ D d)) 2.0) -0.5))))
        (sqrt (/ d h)))
       (if (<= d -1.55e-276)
         (* (* d (- (sqrt (/ (/ 1.0 l) h)))) t_1)
         (*
          (/ d (sqrt l))
          (/ (+ 1.0 (* (pow (* D (/ (* M 0.5) d)) 2.0) t_0)) (sqrt h))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (h / l) * -0.5;
	double t_1 = fma(pow(((M * 0.5) / (d / D)), 2.0), t_0, 1.0);
	double tmp;
	if (d <= -3.8e+133) {
		tmp = t_1 * (-d / sqrt((l * h)));
	} else if (d <= -1.75e-79) {
		tmp = (sqrt((d / l)) * (1.0 + ((h / l) * (pow(((M / 2.0) * (D / d)), 2.0) * -0.5)))) * sqrt((d / h));
	} else if (d <= -1.55e-276) {
		tmp = (d * -sqrt(((1.0 / l) / h))) * t_1;
	} else {
		tmp = (d / sqrt(l)) * ((1.0 + (pow((D * ((M * 0.5) / d)), 2.0) * t_0)) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(h / l) * -0.5)
	t_1 = fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_0, 1.0)
	tmp = 0.0
	if (d <= -3.8e+133)
		tmp = Float64(t_1 * Float64(Float64(-d) / sqrt(Float64(l * h))));
	elseif (d <= -1.75e-79)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(M / 2.0) * Float64(D / d)) ^ 2.0) * -0.5)))) * sqrt(Float64(d / h)));
	elseif (d <= -1.55e-276)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h)))) * t_1);
	else
		tmp = Float64(Float64(d / sqrt(l)) * Float64(Float64(1.0 + Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) * t_0)) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[d, -3.8e+133], N[(t$95$1 * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.75e-79], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(M / 2.0), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.55e-276], N[(N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.8000000000000002e133

   1. Initial program 64.7%

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

   3. Simplified 64.7%

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

      1. frac-times 64.7%

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

      2. associate-/r* 64.7%

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

   6. Applied egg-rr 64.7%

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

      1. expm1-log1p-u 39.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 34.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 24.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 30.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 53.7%

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

      3. associate-*r* 53.7%

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

      4. *-commutative 53.7%

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

      5. *-commutative 53.7%

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

      6. +-commutative 53.7%

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

      7. fma-def 53.7%

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

      8. *-commutative 53.7%

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

      9. associate-/r/ 54.1%

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

      10. *-commutative 54.1%

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

   10. Simplified 54.1%

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

   11. Taylor expanded in d around -inf 80.1%

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

       1. mul-1-neg 80.1%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 80.1%

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

   if -3.8000000000000002e133 < d < -1.75000000000000015e-79

   1. Initial program 87.5%

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

   3. Simplified 87.5%

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

   if -1.75000000000000015e-79 < d < -1.54999999999999995e-276

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

      1. frac-times 49.8%

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

      2. associate-/r* 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. expm1-log1p-u 17.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 4.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 10.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 37.3%

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

      3. associate-*r* 37.3%

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

      4. *-commutative 37.3%

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

      5. *-commutative 37.3%

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

      6. +-commutative 37.3%

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

      7. fma-def 37.3%

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

      8. *-commutative 37.3%

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

      9. associate-/r/ 39.2%

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

      10. *-commutative 39.2%

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

   10. Simplified 39.2%

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

   11. Taylor expanded in d around -inf 68.6%

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

       1. mul-1-neg 68.6%

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

       2. distribute-rgt-neg-in 68.6%

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

       3. *-commutative 68.6%

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

       4. associate-/r* 68.6%

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

   13. Simplified 68.6%

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

   if -1.54999999999999995e-276 < d

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   6. Applied egg-rr 32.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 77.4%

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

      1. fma-udef 77.4%

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

      2. associate-/r* 77.4%

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

      3. div-inv 77.4%

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

      4. metadata-eval 77.4%

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

   10. Applied egg-rr 77.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-79}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-276}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_0, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 2.05 \cdot 10^{+229}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, t_0, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)))
   (if (<= h -1e-309)
     (* (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_0 1.0) (/ (- d) (sqrt (* l h))))
     (if (<= h 2.05e+229)
       (/
        (* d (fma (pow (* D (/ (* M 0.5) d)) 2.0) t_0 1.0))
        (pow (* l h) 0.5))
       (* d (/ (pow l -0.5) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (h / l) * -0.5;
	double tmp;
	if (h <= -1e-309) {
		tmp = fma(pow(((M * 0.5) / (d / D)), 2.0), t_0, 1.0) * (-d / sqrt((l * h)));
	} else if (h <= 2.05e+229) {
		tmp = (d * fma(pow((D * ((M * 0.5) / d)), 2.0), t_0, 1.0)) / pow((l * h), 0.5);
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(h / l) * -0.5)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_0, 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	elseif (h <= 2.05e+229)
		tmp = Float64(Float64(d * fma((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0), t_0, 1.0)) / (Float64(l * h) ^ 0.5));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[h, -1e-309], N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.05e+229], N[(N[(d * N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(l * h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. frac-times 65.9%

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

      2. associate-/r* 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. expm1-log1p-u 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 21.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 23.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 52.7%

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

      3. associate-*r* 52.7%

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

      4. *-commutative 52.7%

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

      5. *-commutative 52.7%

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

      6. +-commutative 52.7%

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

      7. fma-def 52.7%

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

      8. *-commutative 52.7%

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

      9. associate-/r/ 53.6%

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

      10. *-commutative 53.6%

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

   10. Simplified 53.6%

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

   11. Taylor expanded in d around -inf 70.5%

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

       1. mul-1-neg 70.5%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 70.5%

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

   if -1.000000000000002e-309 < h < 2.0500000000000001e229

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   6. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 81.7%

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

      1. frac-times 85.5%

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

      2. associate-/r* 85.5%

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

      3. div-inv 85.5%

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

      4. metadata-eval 85.5%

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

      5. pow1/2 85.5%

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

      6. pow1/2 85.5%

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

      7. pow-prod-down 78.7%

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

   10. Applied egg-rr 78.7%

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

   if 2.0500000000000001e229 < h

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in d around inf 22.4%

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

      1. *-commutative 22.4%

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

      2. associate-/r* 22.4%

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

   7. Simplified 22.4%

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

      1. expm1-log1p-u 22.4%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 3.2%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 3.2%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 55.5%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 55.5%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 55.5%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 2.05 \cdot 10^{+229}:\\ \;\;\;\;\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{t_0}\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+222}:\\ \;\;\;\;\frac{d}{\frac{t_0}{\mathsf{fma}\left({\left(\frac{0.5}{\frac{\frac{d}{D}}{M}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= h -1e-309)
     (*
      (fma (pow (/ (* M 0.5) (/ d D)) 2.0) (* (/ h l) -0.5) 1.0)
      (/ (- d) t_0))
     (if (<= h 6.5e+222)
       (/ d (/ t_0 (fma (pow (/ 0.5 (/ (/ d D) M)) 2.0) (/ (* h -0.5) l) 1.0)))
       (* d (/ (pow l -0.5) (sqrt h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (h <= -1e-309) {
		tmp = fma(pow(((M * 0.5) / (d / D)), 2.0), ((h / l) * -0.5), 1.0) * (-d / t_0);
	} else if (h <= 6.5e+222) {
		tmp = d / (t_0 / fma(pow((0.5 / ((d / D) / M)), 2.0), ((h * -0.5) / l), 1.0));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), Float64(Float64(h / l) * -0.5), 1.0) * Float64(Float64(-d) / t_0));
	elseif (h <= 6.5e+222)
		tmp = Float64(d / Float64(t_0 / fma((Float64(0.5 / Float64(Float64(d / D) / M)) ^ 2.0), Float64(Float64(h * -0.5) / l), 1.0)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1e-309], N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-d) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 6.5e+222], N[(d / N[(t$95$0 / N[(N[Power[N[(0.5 / N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h * -0.5), $MachinePrecision] / l), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. frac-times 65.9%

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

      2. associate-/r* 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. expm1-log1p-u 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 21.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 23.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 52.7%

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

      3. associate-*r* 52.7%

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

      4. *-commutative 52.7%

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

      5. *-commutative 52.7%

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

      6. +-commutative 52.7%

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

      7. fma-def 52.7%

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

      8. *-commutative 52.7%

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

      9. associate-/r/ 53.6%

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

      10. *-commutative 53.6%

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

   10. Simplified 53.6%

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

   11. Taylor expanded in d around -inf 70.5%

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

       1. mul-1-neg 70.5%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 70.5%

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

   if -1.000000000000002e-309 < h < 6.5000000000000006e222

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   6. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 81.7%

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

      1. expm1-log1p-u 50.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{\ell}} \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\right)\right)} \]

      2. expm1-udef 36.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell}} \cdot \frac{\mathsf{fma}\left({\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{\sqrt{h}}\right)} - 1} \]

   10. Applied egg-rr 32.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 46.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d \cdot \mathsf{fma}\left({\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)}{{\left(\ell \cdot h\right)}^{0.5}}\right)\right)} \]

       2. expm1-log1p 78.7%

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

       3. associate-/l* 77.8%

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

       4. unpow1/2 77.8%

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

       5. *-commutative 77.8%

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

       6. *-commutative 77.8%

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

       7. associate-/r/ 77.4%

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

       8. *-commutative 77.4%

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

       9. associate-/l* 77.5%

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

       10. associate-*l/ 77.5%

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

   12. Simplified 77.5%

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

   if 6.5000000000000006e222 < h

   1. Initial program 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in d around inf 22.4%

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

      1. *-commutative 22.4%

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

      2. associate-/r* 22.4%

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

   7. Simplified 22.4%

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

      1. expm1-log1p-u 22.4%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 3.2%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 3.2%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 3.2%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 55.5%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 55.5%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 55.5%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{elif}\;h \leq 6.5 \cdot 10^{+222}:\\ \;\;\;\;\frac{d}{\frac{\sqrt{\ell \cdot h}}{\mathsf{fma}\left({\left(\frac{0.5}{\frac{\frac{d}{D}}{M}}\right)}^{2}, \frac{h \cdot -0.5}{\ell}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot t_0}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)))
   (if (<= h -1e-309)
     (*
      (* d (- (sqrt (/ (/ 1.0 l) h))))
      (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_0 1.0))
     (*
      (/ d (sqrt l))
      (/ (+ 1.0 (* (pow (* D (/ (* M 0.5) d)) 2.0) t_0)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (h / l) * -0.5;
	double tmp;
	if (h <= -1e-309) {
		tmp = (d * -sqrt(((1.0 / l) / h))) * fma(pow(((M * 0.5) / (d / D)), 2.0), t_0, 1.0);
	} else {
		tmp = (d / sqrt(l)) * ((1.0 + (pow((D * ((M * 0.5) / d)), 2.0) * t_0)) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(h / l) * -0.5)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(Float64(d * Float64(-sqrt(Float64(Float64(1.0 / l) / h)))) * fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_0, 1.0));
	else
		tmp = Float64(Float64(d / sqrt(l)) * Float64(Float64(1.0 + Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) * t_0)) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[h, -1e-309], N[(N[(d * (-N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. frac-times 65.9%

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

      2. associate-/r* 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. expm1-log1p-u 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 21.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 23.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 52.7%

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

      3. associate-*r* 52.7%

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

      4. *-commutative 52.7%

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

      5. *-commutative 52.7%

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

      6. +-commutative 52.7%

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

      7. fma-def 52.7%

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

      8. *-commutative 52.7%

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

      9. associate-/r/ 53.6%

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

      10. *-commutative 53.6%

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

   10. Simplified 53.6%

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

   11. Taylor expanded in d around -inf 70.5%

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

       1. mul-1-neg 70.5%

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

       2. distribute-rgt-neg-in 70.5%

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

       3. *-commutative 70.5%

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

       4. associate-/r* 71.2%

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

   13. Simplified 71.2%

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

   if -1.000000000000002e-309 < h

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   6. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 78.1%

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

      1. fma-udef 78.1%

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

      2. associate-/r* 78.1%

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

      3. div-inv 78.1%

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

      4. metadata-eval 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(d \cdot \left(-\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right) \cdot \mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{h}{\ell} \cdot -0.5\\ \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, t_0, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot t_0}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* (/ h l) -0.5)))
   (if (<= h -1e-309)
     (* (fma (pow (/ (* M 0.5) (/ d D)) 2.0) t_0 1.0) (/ (- d) (sqrt (* l h))))
     (*
      (/ d (sqrt l))
      (/ (+ 1.0 (* (pow (* D (/ (* M 0.5) d)) 2.0) t_0)) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (h / l) * -0.5;
	double tmp;
	if (h <= -1e-309) {
		tmp = fma(pow(((M * 0.5) / (d / D)), 2.0), t_0, 1.0) * (-d / sqrt((l * h)));
	} else {
		tmp = (d / sqrt(l)) * ((1.0 + (pow((D * ((M * 0.5) / d)), 2.0) * t_0)) / sqrt(h));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(h / l) * -0.5)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = Float64(fma((Float64(Float64(M * 0.5) / Float64(d / D)) ^ 2.0), t_0, 1.0) * Float64(Float64(-d) / sqrt(Float64(l * h))));
	else
		tmp = Float64(Float64(d / sqrt(l)) * Float64(Float64(1.0 + Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) * t_0)) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[h, -1e-309], N[(N[(N[Power[N[(N[(M * 0.5), $MachinePrecision] / N[(d / D), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] * N[((-d) / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

      1. frac-times 65.9%

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

      2. associate-/r* 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. expm1-log1p-u 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 21.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 23.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 52.7%

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

      3. associate-*r* 52.7%

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

      4. *-commutative 52.7%

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

      5. *-commutative 52.7%

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

      6. +-commutative 52.7%

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

      7. fma-def 52.7%

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

      8. *-commutative 52.7%

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

      9. associate-/r/ 53.6%

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

      10. *-commutative 53.6%

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

   10. Simplified 53.6%

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

   11. Taylor expanded in d around -inf 70.5%

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

       1. mul-1-neg 70.5%

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

       2. rem-exp-log 0.0%

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

       3. unpow1/2 0.0%

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

       4. rem-exp-log 0.0%

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

       5. rec-exp 0.0%

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

       6. exp-prod 0.0%

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

       7. distribute-lft-neg-in 0.0%

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

       8. *-commutative 0.0%

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

       9. distribute-lft-neg-in 0.0%

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

       10. metadata-eval 0.0%

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

       11. log-pow 0.0%

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

       12. exp-sum 0.0%

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

       13. log-pow 0.0%

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

       14. metadata-eval 0.0%

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

       15. distribute-lft-neg-in 0.0%

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

       16. log-pow 0.0%

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

       17. unpow1/2 0.0%

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

       18. sub-neg 0.0%

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

   13. Simplified 70.5%

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

   if -1.000000000000002e-309 < h

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   6. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 78.1%

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

      1. fma-udef 78.1%

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

      2. associate-/r* 78.1%

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

      3. div-inv 78.1%

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

      4. metadata-eval 78.1%

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{M \cdot 0.5}{\frac{d}{D}}\right)}^{2}, \frac{h}{\ell} \cdot -0.5, 1\right) \cdot \frac{-d}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \frac{1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+182}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-281}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{+127}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<= d -2.4e+182)
     (fabs t_0)
     (if (<= d -4e-220)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (+ 1.0 (* (pow (* D (/ (* M 0.5) d)) 2.0) (* (/ h l) -0.5))))
       (if (<= d 1.05e-281)
         (* d (log (exp (pow (* l h) -0.5))))
         (if (<= d 1.1e+127)
           (* t_0 (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (/ D 2.0) d)) 2.0)))))
           (* d (/ (pow l -0.5) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (d <= -2.4e+182) {
		tmp = fabs(t_0);
	} else if (d <= -4e-220) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= 1.05e-281) {
		tmp = d * log(exp(pow((l * h), -0.5)));
	} else if (d <= 1.1e+127) {
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * pow((M * ((D / 2.0) / d)), 2.0))));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d / sqrt((l * h))
    if (d <= (-2.4d+182)) then
        tmp = abs(t_0)
    else if (d <= (-4d-220)) then
        tmp = sqrt(((d / l) * (d / h))) * (1.0d0 + (((d_1 * ((m * 0.5d0) / d)) ** 2.0d0) * ((h / l) * (-0.5d0))))
    else if (d <= 1.05d-281) then
        tmp = d * log(exp(((l * h) ** (-0.5d0))))
    else if (d <= 1.1d+127) then
        tmp = t_0 * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * ((d_1 / 2.0d0) / d)) ** 2.0d0))))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (d <= -2.4e+182) {
		tmp = Math.abs(t_0);
	} else if (d <= -4e-220) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (Math.pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= 1.05e-281) {
		tmp = d * Math.log(Math.exp(Math.pow((l * h), -0.5)));
	} else if (d <= 1.1e+127) {
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D / 2.0) / d)), 2.0))));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if d <= -2.4e+182:
		tmp = math.fabs(t_0)
	elif d <= -4e-220:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (math.pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)))
	elif d <= 1.05e-281:
		tmp = d * math.log(math.exp(math.pow((l * h), -0.5)))
	elif d <= 1.1e+127:
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D / 2.0) / d)), 2.0))))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (d <= -2.4e+182)
		tmp = abs(t_0);
	elseif (d <= -4e-220)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (d <= 1.05e-281)
		tmp = Float64(d * log(exp((Float64(l * h) ^ -0.5))));
	elseif (d <= 1.1e+127)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D / 2.0) / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = d / sqrt((l * h));
	tmp = 0.0;
	if (d <= -2.4e+182)
		tmp = abs(t_0);
	elseif (d <= -4e-220)
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (((D * ((M * 0.5) / d)) ^ 2.0) * ((h / l) * -0.5)));
	elseif (d <= 1.05e-281)
		tmp = d * log(exp(((l * h) ^ -0.5)));
	elseif (d <= 1.1e+127)
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * ((M * ((D / 2.0) / d)) ^ 2.0))));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.4e+182], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -4e-220], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-281], N[(d * N[Log[N[Exp[N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.1e+127], N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.4000000000000001e182

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in d around inf 9.5%

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

      1. *-commutative 9.5%

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

      2. associate-/r* 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in d around 0 9.5%

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

      1. unpow1/2 9.5%

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

      2. rem-exp-log 9.5%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 9.5%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 9.5%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 9.5%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 9.5%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 9.5%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 9.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 9.5%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-unprod 19.4%

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

       3. pow2 19.4%

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

   12. Applied egg-rr 19.4%

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

       1. unpow2 19.4%

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

       2. rem-sqrt-square 64.2%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       3. rem-exp-log 0.4%

          \[\leadsto \left|\color{blue}{e^{\log \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

       4. log-prod 0.0%

          \[\leadsto \left|e^{\color{blue}{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

       5. log-pow 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}}\right| \]

       6. metadata-eval 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5\right)} \cdot \log \left(h \cdot \ell\right)}\right| \]

       7. distribute-lft-neg-in 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}}\right| \]

       8. log-pow 0.0%

          \[\leadsto \left|e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)}\right| \]

       9. unpow1/2 0.0%

          \[\leadsto \left|e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)}\right| \]

       10. sub-neg 0.0%

           \[\leadsto \left|e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}}\right| \]

       11. log-div 0.4%

           \[\leadsto \left|e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}}\right| \]

       12. rem-exp-log 64.2%

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

   14. Simplified 64.2%

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

   if -2.4000000000000001e182 < d < -3.99999999999999997e-220

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

      1. frac-times 74.7%

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

      2. associate-/r* 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. expm1-log1p-u 35.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 20.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 26.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 62.5%

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

      3. associate-*r* 62.5%

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

      4. *-commutative 62.5%

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

      5. *-commutative 62.5%

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

      6. +-commutative 62.5%

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

      7. fma-def 62.5%

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

      8. *-commutative 62.5%

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

      9. associate-/r/ 63.7%

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

      10. *-commutative 63.7%

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

   10. Simplified 63.7%

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

       1. fma-udef 63.7%

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

       2. associate-/r/ 62.5%

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

       3. *-commutative 62.5%

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

   12. Applied egg-rr 62.5%

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

   if -3.99999999999999997e-220 < d < 1.0499999999999999e-281

   1. Initial program 26.9%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in d around inf 11.4%

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

      1. *-commutative 11.4%

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

      2. associate-/r* 11.4%

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

   7. Simplified 11.4%

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

   8. Taylor expanded in d around 0 11.4%

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

      1. unpow1/2 11.4%

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

      2. rem-exp-log 11.4%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 11.4%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 11.4%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 11.4%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 11.4%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 11.4%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 11.4%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 11.4%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-log-exp 37.4%

          \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   12. Applied egg-rr 37.4%

       \[\leadsto d \cdot \color{blue}{\log \left(e^{{\left(h \cdot \ell\right)}^{-0.5}}\right)} \]

   if 1.0499999999999999e-281 < d < 1.1000000000000001e127

   1. Initial program 63.6%

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

   3. Simplified 62.3%

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

      1. clear-num 62.3%

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

      2. frac-times 63.5%

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

      3. *-un-lft-identity 63.5%

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

      4. associate-*l/ 63.5%

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

      5. *-un-lft-identity 63.5%

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

      6. times-frac 63.5%

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

      7. metadata-eval 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. expm1-log1p-u 32.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 14.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + -0.5 \cdot \left({\left(\frac{\frac{D}{2}}{\frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 31.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + -0.5 \cdot \left({\left(\frac{\frac{D}{2}}{\frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 68.6%

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

      3. associate-/r/ 67.4%

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

   10. Simplified 67.4%

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

   if 1.1000000000000001e127 < d

   1. Initial program 74.7%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in d around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-/r* 62.6%

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

   7. Simplified 62.6%

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

      1. expm1-log1p-u 61.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 42.5%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 44.6%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 82.6%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 83.9%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 83.9%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+182}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-281}:\\ \;\;\;\;d \cdot \log \left(e^{{\left(\ell \cdot h\right)}^{-0.5}}\right)\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{+181}:\\ \;\;\;\;\left|t_0\right|\\ \mathbf{elif}\;d \leq -1.46 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-281}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(\ell \cdot h + -1\right)\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+127}:\\ \;\;\;\;t_0 \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))))
   (if (<= d -2.95e+181)
     (fabs t_0)
     (if (<= d -1.46e-215)
       (*
        (sqrt (* (/ d l) (/ d h)))
        (+ 1.0 (* (pow (* D (/ (* M 0.5) d)) 2.0) (* (/ h l) -0.5))))
       (if (<= d 1.05e-281)
         (* d (exp (* 0.5 (- (log1p (+ (* l h) -1.0))))))
         (if (<= d 2e+127)
           (* t_0 (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (/ D 2.0) d)) 2.0)))))
           (* d (/ (pow l -0.5) (sqrt h)))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double tmp;
	if (d <= -2.95e+181) {
		tmp = fabs(t_0);
	} else if (d <= -1.46e-215) {
		tmp = sqrt(((d / l) * (d / h))) * (1.0 + (pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= 1.05e-281) {
		tmp = d * exp((0.5 * -log1p(((l * h) + -1.0))));
	} else if (d <= 2e+127) {
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * pow((M * ((D / 2.0) / d)), 2.0))));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double tmp;
	if (d <= -2.95e+181) {
		tmp = Math.abs(t_0);
	} else if (d <= -1.46e-215) {
		tmp = Math.sqrt(((d / l) * (d / h))) * (1.0 + (Math.pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)));
	} else if (d <= 1.05e-281) {
		tmp = d * Math.exp((0.5 * -Math.log1p(((l * h) + -1.0))));
	} else if (d <= 2e+127) {
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D / 2.0) / d)), 2.0))));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = d / math.sqrt((l * h))
	tmp = 0
	if d <= -2.95e+181:
		tmp = math.fabs(t_0)
	elif d <= -1.46e-215:
		tmp = math.sqrt(((d / l) * (d / h))) * (1.0 + (math.pow((D * ((M * 0.5) / d)), 2.0) * ((h / l) * -0.5)))
	elif d <= 1.05e-281:
		tmp = d * math.exp((0.5 * -math.log1p(((l * h) + -1.0))))
	elif d <= 2e+127:
		tmp = t_0 * (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D / 2.0) / d)), 2.0))))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	tmp = 0.0
	if (d <= -2.95e+181)
		tmp = abs(t_0);
	elseif (d <= -1.46e-215)
		tmp = Float64(sqrt(Float64(Float64(d / l) * Float64(d / h))) * Float64(1.0 + Float64((Float64(D * Float64(Float64(M * 0.5) / d)) ^ 2.0) * Float64(Float64(h / l) * -0.5))));
	elseif (d <= 1.05e-281)
		tmp = Float64(d * exp(Float64(0.5 * Float64(-log1p(Float64(Float64(l * h) + -1.0))))));
	elseif (d <= 2e+127)
		tmp = Float64(t_0 * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D / 2.0) / d)) ^ 2.0)))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.95e+181], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[d, -1.46e-215], N[(N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[N[(D * N[(N[(M * 0.5), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-281], N[(d * N[Exp[N[(0.5 * (-N[Log[1 + N[(N[(l * h), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e+127], N[(t$95$0 * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.9499999999999999e181

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in d around inf 9.5%

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

      1. *-commutative 9.5%

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

      2. associate-/r* 9.5%

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

   7. Simplified 9.5%

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

   8. Taylor expanded in d around 0 9.5%

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

      1. unpow1/2 9.5%

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

      2. rem-exp-log 9.5%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 9.5%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 9.5%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 9.5%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 9.5%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 9.5%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 9.5%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 9.5%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-unprod 19.4%

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

       3. pow2 19.4%

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

   12. Applied egg-rr 19.4%

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

       1. unpow2 19.4%

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

       2. rem-sqrt-square 64.2%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

       3. rem-exp-log 0.4%

          \[\leadsto \left|\color{blue}{e^{\log \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

       4. log-prod 0.0%

          \[\leadsto \left|e^{\color{blue}{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

       5. log-pow 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}}\right| \]

       6. metadata-eval 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5\right)} \cdot \log \left(h \cdot \ell\right)}\right| \]

       7. distribute-lft-neg-in 0.0%

          \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}}\right| \]

       8. log-pow 0.0%

          \[\leadsto \left|e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)}\right| \]

       9. unpow1/2 0.0%

          \[\leadsto \left|e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)}\right| \]

       10. sub-neg 0.0%

           \[\leadsto \left|e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}}\right| \]

       11. log-div 0.4%

           \[\leadsto \left|e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}}\right| \]

       12. rem-exp-log 64.2%

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

   14. Simplified 64.2%

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

   if -2.9499999999999999e181 < d < -1.4600000000000001e-215

   1. Initial program 74.7%

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

   3. Simplified 74.7%

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

      1. frac-times 74.7%

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

      2. associate-/r* 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. expm1-log1p-u 35.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 20.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 17.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 26.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 62.5%

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

      3. associate-*r* 62.5%

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

      4. *-commutative 62.5%

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

      5. *-commutative 62.5%

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

      6. +-commutative 62.5%

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

      7. fma-def 62.5%

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

      8. *-commutative 62.5%

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

      9. associate-/r/ 63.7%

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

      10. *-commutative 63.7%

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

   10. Simplified 63.7%

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

       1. fma-udef 63.7%

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

       2. associate-/r/ 62.5%

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

       3. *-commutative 62.5%

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

   12. Applied egg-rr 62.5%

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

   if -1.4600000000000001e-215 < d < 1.0499999999999999e-281

   1. Initial program 26.9%

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

   3. Simplified 27.1%

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

   5. Taylor expanded in d around inf 11.4%

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

      1. *-commutative 11.4%

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

      2. associate-/r* 11.4%

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

   7. Simplified 11.4%

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

      1. pow1/2 11.4%

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

      2. pow-to-exp 11.4%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{\frac{1}{\ell}}{h}\right) \cdot 0.5}} \]

      3. associate-/l/ 11.4%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      4. log-rec 11.4%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   9. Applied egg-rr 11.4%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 33.2%

          \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5} \]

       2. expm1-udef 33.2%

          \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5} \]

       3. add-exp-log 33.2%

          \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{h \cdot \ell} - 1\right)\right) \cdot 0.5} \]

   11. Applied egg-rr 33.2%

       \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)}\right) \cdot 0.5} \]

   if 1.0499999999999999e-281 < d < 1.99999999999999991e127

   1. Initial program 63.6%

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

   3. Simplified 62.3%

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

      1. clear-num 62.3%

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

      2. frac-times 63.5%

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

      3. *-un-lft-identity 63.5%

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

      4. associate-*l/ 63.5%

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

      5. *-un-lft-identity 63.5%

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

      6. times-frac 63.5%

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

      7. metadata-eval 63.5%

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

   6. Applied egg-rr 63.5%

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

      1. expm1-log1p-u 32.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 14.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 14.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + -0.5 \cdot \left({\left(\frac{\frac{D}{2}}{\frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 31.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + -0.5 \cdot \left({\left(\frac{\frac{D}{2}}{\frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 68.6%

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

      3. associate-/r/ 67.4%

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

   10. Simplified 67.4%

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

   if 1.99999999999999991e127 < d

   1. Initial program 74.7%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in d around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-/r* 62.6%

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

   7. Simplified 62.6%

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

      1. expm1-log1p-u 61.5%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 42.5%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 44.6%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 44.6%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 82.6%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 83.9%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 83.9%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.95 \cdot 10^{+181}:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}}\right|\\ \mathbf{elif}\;d \leq -1.46 \cdot 10^{-215}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}} \cdot \left(1 + {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2} \cdot \left(\frac{h}{\ell} \cdot -0.5\right)\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-281}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(\ell \cdot h + -1\right)\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.42 \cdot 10^{-71}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(\ell \cdot h + -1\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.1 \cdot 10^{+245}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \sqrt{\frac{1}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.42e-71)
   (fabs (* d (pow (* l h) -0.5)))
   (if (<= l -5e-310)
     (* d (exp (* 0.5 (- (log1p (+ (* l h) -1.0))))))
     (if (<= l 7.1e+245)
       (*
        (/ d (sqrt (* l h)))
        (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (/ (/ D 2.0) d)) 2.0)))))
       (* (/ d (sqrt l)) (sqrt (/ 1.0 h)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.42e-71) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else if (l <= -5e-310) {
		tmp = d * exp((0.5 * -log1p(((l * h) + -1.0))));
	} else if (l <= 7.1e+245) {
		tmp = (d / sqrt((l * h))) * (1.0 + (-0.5 * ((h / l) * pow((M * ((D / 2.0) / d)), 2.0))));
	} else {
		tmp = (d / sqrt(l)) * sqrt((1.0 / h));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.42e-71) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else if (l <= -5e-310) {
		tmp = d * Math.exp((0.5 * -Math.log1p(((l * h) + -1.0))));
	} else if (l <= 7.1e+245) {
		tmp = (d / Math.sqrt((l * h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * ((D / 2.0) / d)), 2.0))));
	} else {
		tmp = (d / Math.sqrt(l)) * Math.sqrt((1.0 / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.42e-71:
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	elif l <= -5e-310:
		tmp = d * math.exp((0.5 * -math.log1p(((l * h) + -1.0))))
	elif l <= 7.1e+245:
		tmp = (d / math.sqrt((l * h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * ((D / 2.0) / d)), 2.0))))
	else:
		tmp = (d / math.sqrt(l)) * math.sqrt((1.0 / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.42e-71)
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (l <= -5e-310)
		tmp = Float64(d * exp(Float64(0.5 * Float64(-log1p(Float64(Float64(l * h) + -1.0))))));
	elseif (l <= 7.1e+245)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(Float64(D / 2.0) / d)) ^ 2.0)))));
	else
		tmp = Float64(Float64(d / sqrt(l)) * sqrt(Float64(1.0 / h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.42e-71], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Exp[N[(0.5 * (-N[Log[1 + N[(N[(l * h), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.1e+245], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(N[(D / 2.0), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.4199999999999999e-71

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in d around inf 3.8%

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

      1. *-commutative 3.8%

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

      2. associate-/r* 3.8%

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

   7. Simplified 3.8%

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

   8. Taylor expanded in d around 0 3.8%

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

      1. unpow1/2 3.8%

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

      2. rem-exp-log 3.8%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 3.8%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 3.8%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 3.8%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 3.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 3.8%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 3.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 3.8%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 1.9%

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

       2. sqrt-unprod 28.5%

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

       3. pow2 28.5%

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

   12. Applied egg-rr 28.5%

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

       1. unpow2 28.5%

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

       2. rem-sqrt-square 46.6%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   14. Simplified 46.6%

       \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   if -1.4199999999999999e-71 < l < -4.999999999999985e-310

   1. Initial program 68.8%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in d around inf 16.5%

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

      1. *-commutative 16.5%

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

      2. associate-/r* 16.5%

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

   7. Simplified 16.5%

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

      1. pow1/2 16.5%

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

      2. pow-to-exp 16.5%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{\frac{1}{\ell}}{h}\right) \cdot 0.5}} \]

      3. associate-/l/ 16.5%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      4. log-rec 16.5%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   9. Applied egg-rr 16.5%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 44.1%

          \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5} \]

       2. expm1-udef 44.1%

          \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5} \]

       3. add-exp-log 44.1%

          \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{h \cdot \ell} - 1\right)\right) \cdot 0.5} \]

   11. Applied egg-rr 44.1%

       \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)}\right) \cdot 0.5} \]

   if -4.999999999999985e-310 < l < 7.09999999999999957e245

   1. Initial program 66.7%

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

   3. Simplified 65.7%

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

      1. clear-num 65.7%

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

      2. frac-times 66.6%

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

      3. *-un-lft-identity 66.6%

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

      4. associate-*l/ 66.6%

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

      5. *-un-lft-identity 66.6%

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

      6. times-frac 66.6%

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

      7. metadata-eval 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. expm1-log1p-u 38.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 26.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 26.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 + -0.5 \cdot \left({\left(\frac{\frac{D}{2}}{\frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 36.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + -0.5 \cdot \left({\left(\frac{\frac{D}{2}}{\frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 69.3%

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

      3. associate-/r/ 68.4%

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

   10. Simplified 68.4%

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

   if 7.09999999999999957e245 < l

   1. Initial program 60.3%

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

   3. Simplified 66.9%

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

   6. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2}\right)\right)\right)} - 1} \]

   8. Simplified 79.5%

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

   9. Taylor expanded in D around 0 79.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.42 \cdot 10^{-71}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(\ell \cdot h + -1\right)\right)}\\ \mathbf{elif}\;\ell \leq 7.1 \cdot 10^{+245}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{\frac{D}{2}}{d}\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell}} \cdot \sqrt{\frac{1}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-71}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(\ell \cdot h + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.7e-71)
   (fabs (* d (pow (* l h) -0.5)))
   (if (<= l -5e-310)
     (* d (exp (* 0.5 (- (log1p (+ (* l h) -1.0))))))
     (* d (/ (pow l -0.5) (sqrt h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-71) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else if (l <= -5e-310) {
		tmp = d * exp((0.5 * -log1p(((l * h) + -1.0))));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-71) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else if (l <= -5e-310) {
		tmp = d * Math.exp((0.5 * -Math.log1p(((l * h) + -1.0))));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.7e-71:
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	elif l <= -5e-310:
		tmp = d * math.exp((0.5 * -math.log1p(((l * h) + -1.0))))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.7e-71)
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	elseif (l <= -5e-310)
		tmp = Float64(d * exp(Float64(0.5 * Float64(-log1p(Float64(Float64(l * h) + -1.0))))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.7e-71], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5e-310], N[(d * N[Exp[N[(0.5 * (-N[Log[1 + N[(N[(l * h), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.70000000000000002e-71

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in d around inf 3.8%

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

      1. *-commutative 3.8%

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

      2. associate-/r* 3.8%

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

   7. Simplified 3.8%

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

   8. Taylor expanded in d around 0 3.8%

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

      1. unpow1/2 3.8%

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

      2. rem-exp-log 3.8%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 3.8%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 3.8%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 3.8%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 3.8%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 3.8%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 3.8%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 3.8%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 1.9%

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

       2. sqrt-unprod 28.5%

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

       3. pow2 28.5%

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

   12. Applied egg-rr 28.5%

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

       1. unpow2 28.5%

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

       2. rem-sqrt-square 46.6%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   14. Simplified 46.6%

       \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   if -1.70000000000000002e-71 < l < -4.999999999999985e-310

   1. Initial program 68.8%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in d around inf 16.5%

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

      1. *-commutative 16.5%

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

      2. associate-/r* 16.5%

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

   7. Simplified 16.5%

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

      1. pow1/2 16.5%

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

      2. pow-to-exp 16.5%

         \[\leadsto d \cdot \color{blue}{e^{\log \left(\frac{\frac{1}{\ell}}{h}\right) \cdot 0.5}} \]

      3. associate-/l/ 16.5%

         \[\leadsto d \cdot e^{\log \color{blue}{\left(\frac{1}{h \cdot \ell}\right)} \cdot 0.5} \]

      4. log-rec 16.5%

         \[\leadsto d \cdot e^{\color{blue}{\left(-\log \left(h \cdot \ell\right)\right)} \cdot 0.5} \]

   9. Applied egg-rr 16.5%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]
   10. Step-by-step derivation

       1. log1p-expm1-u 44.1%

          \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(h \cdot \ell\right)\right)\right)}\right) \cdot 0.5} \]

       2. expm1-udef 44.1%

          \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{e^{\log \left(h \cdot \ell\right)} - 1}\right)\right) \cdot 0.5} \]

       3. add-exp-log 44.1%

          \[\leadsto d \cdot e^{\left(-\mathsf{log1p}\left(\color{blue}{h \cdot \ell} - 1\right)\right) \cdot 0.5} \]

   11. Applied egg-rr 44.1%

       \[\leadsto d \cdot e^{\left(-\color{blue}{\mathsf{log1p}\left(h \cdot \ell - 1\right)}\right) \cdot 0.5} \]

   if -4.999999999999985e-310 < l

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in d around inf 40.4%

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

      1. *-commutative 40.4%

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

      2. associate-/r* 40.8%

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

   7. Simplified 40.8%

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

      1. expm1-log1p-u 39.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 25.6%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 28.4%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 50.3%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 51.5%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 51.5%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-71}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;d \cdot e^{0.5 \cdot \left(-\mathsf{log1p}\left(\ell \cdot h + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= h -1e-309)
   (fabs (* d (pow (* l h) -0.5)))
   (* d (/ (pow l -0.5) (sqrt h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-309) {
		tmp = fabs((d * pow((l * h), -0.5)));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (h <= (-1d-309)) then
        tmp = abs((d * ((l * h) ** (-0.5d0))))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (h <= -1e-309) {
		tmp = Math.abs((d * Math.pow((l * h), -0.5)));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if h <= -1e-309:
		tmp = math.fabs((d * math.pow((l * h), -0.5)))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (h <= -1e-309)
		tmp = abs(Float64(d * (Float64(l * h) ^ -0.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (h <= -1e-309)
		tmp = abs((d * ((l * h) ^ -0.5)));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[h, -1e-309], N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < -1.000000000000002e-309

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in d around inf 8.9%

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

      1. *-commutative 8.9%

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

      2. associate-/r* 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in d around 0 8.9%

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

      1. unpow1/2 8.9%

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

      2. rem-exp-log 8.9%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 8.9%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 8.9%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 8.9%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 8.9%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 8.9%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 8.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 8.9%

       \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 1.2%

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

       2. sqrt-unprod 24.3%

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

       3. pow2 24.3%

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

   12. Applied egg-rr 24.3%

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

       1. unpow2 24.3%

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

       2. rem-sqrt-square 37.6%

          \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   14. Simplified 37.6%

       \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

   if -1.000000000000002e-309 < h

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in d around inf 40.4%

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

      1. *-commutative 40.4%

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

      2. associate-/r* 40.8%

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

   7. Simplified 40.8%

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

      1. expm1-log1p-u 39.9%

         \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 25.6%

         \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

      3. sqrt-div 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}}\right)} - 1\right) \]

      4. pow1/2 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\left(\frac{1}{\ell}\right)}^{0.5}}}{\sqrt{h}}\right)} - 1\right) \]

      5. inv-pow 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\color{blue}{\left({\ell}^{-1}\right)}}^{0.5}}{\sqrt{h}}\right)} - 1\right) \]

      6. pow-pow 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\ell}^{\left(-1 \cdot 0.5\right)}}}{\sqrt{h}}\right)} - 1\right) \]

      7. metadata-eval 28.4%

         \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}}\right)} - 1\right) \]

   9. Applied egg-rr 28.4%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 50.3%

          \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{-0.5}}{\sqrt{h}}\right)\right)} \]

       2. expm1-log1p 51.5%

          \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]

   11. Simplified 51.5%

       \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right| \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (fabs (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	return fabs((d * pow((l * h), -0.5)));
}

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 = abs((d * ((l * h) ** (-0.5d0))))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return Math.abs((d * Math.pow((l * h), -0.5)));
}

Python

def code(d, h, l, M, D):
	return math.fabs((d * math.pow((l * h), -0.5)))

Julia

function code(d, h, l, M, D)
	return abs(Float64(d * (Float64(l * h) ^ -0.5)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = abs((d * ((l * h) ^ -0.5)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[Abs[N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 65.9%

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

3. Simplified 65.9%

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

5. Taylor expanded in d around inf 23.3%

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

   1. *-commutative 23.3%

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

   2. associate-/r* 23.5%

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

7. Simplified 23.5%

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

8. Taylor expanded in d around 0 23.3%

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

   1. unpow1/2 23.3%

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

   2. rem-exp-log 22.4%

      \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

   3. exp-neg 22.4%

      \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

   4. exp-prod 22.7%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

   5. distribute-lft-neg-out 22.7%

      \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

   6. distribute-rgt-neg-in 22.7%

      \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

   7. metadata-eval 22.7%

      \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

   8. exp-to-pow 23.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Simplified 23.6%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 19.4%

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

    2. sqrt-unprod 27.1%

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

    3. pow2 27.1%

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

12. Applied egg-rr 27.1%

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

    1. unpow2 27.1%

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

    2. rem-sqrt-square 39.2%

       \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

14. Simplified 39.2%

    \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

15. Final simplification 39.2%

    \[\leadsto \left|d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right| \]
16. Add Preprocessing

Alternative 18: 42.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (fabs (/ d (sqrt (* l h)))))

C

double code(double d, double h, double l, double M, double D) {
	return fabs((d / sqrt((l * h))));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = abs((d / sqrt((l * h))))
end function

Java

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

Python

def code(d, h, l, M, D):
	return math.fabs((d / math.sqrt((l * h))))

Julia

function code(d, h, l, M, D)
	return abs(Float64(d / sqrt(Float64(l * h))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = abs((d / sqrt((l * h))));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[Abs[N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 65.9%

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

3. Simplified 65.9%

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

5. Taylor expanded in d around inf 23.3%

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

   1. *-commutative 23.3%

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

   2. associate-/r* 23.5%

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

7. Simplified 23.5%

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

8. Taylor expanded in d around 0 23.3%

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

   1. unpow1/2 23.3%

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

   2. rem-exp-log 22.4%

      \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

   3. exp-neg 22.4%

      \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

   4. exp-prod 22.7%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

   5. distribute-lft-neg-out 22.7%

      \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

   6. distribute-rgt-neg-in 22.7%

      \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

   7. metadata-eval 22.7%

      \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

   8. exp-to-pow 23.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Simplified 23.6%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 19.4%

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

    2. sqrt-unprod 27.1%

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

    3. pow2 27.1%

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

12. Applied egg-rr 27.1%

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

    1. unpow2 27.1%

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

    2. rem-sqrt-square 39.2%

       \[\leadsto \color{blue}{\left|d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right|} \]

    3. rem-exp-log 18.5%

       \[\leadsto \left|\color{blue}{e^{\log \left(d \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

    4. log-prod 17.6%

       \[\leadsto \left|e^{\color{blue}{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}}\right| \]

    5. log-pow 17.6%

       \[\leadsto \left|e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}}\right| \]

    6. metadata-eval 17.6%

       \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5\right)} \cdot \log \left(h \cdot \ell\right)}\right| \]

    7. distribute-lft-neg-in 17.6%

       \[\leadsto \left|e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}}\right| \]

    8. log-pow 17.6%

       \[\leadsto \left|e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)}\right| \]

    9. unpow1/2 17.6%

       \[\leadsto \left|e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)}\right| \]

    10. sub-neg 17.6%

        \[\leadsto \left|e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}}\right| \]

    11. log-div 18.5%

        \[\leadsto \left|e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}}\right| \]

    12. rem-exp-log 39.2%

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

14. Simplified 39.2%

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

15. Final simplification 39.2%

    \[\leadsto \left|\frac{d}{\sqrt{\ell \cdot h}}\right| \]
16. Add Preprocessing

Alternative 19: 37.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.25 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.25e-172) (sqrt (* (/ d l) (/ d h))) (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.25e-172) {
		tmp = sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * pow((l * h), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-2.25d-172)) then
        tmp = sqrt(((d / l) * (d / h)))
    else
        tmp = d * ((l * h) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.25e-172) {
		tmp = Math.sqrt(((d / l) * (d / h)));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.25e-172:
		tmp = math.sqrt(((d / l) * (d / h)))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.25e-172)
		tmp = sqrt(Float64(Float64(d / l) * Float64(d / h)));
	else
		tmp = Float64(d * (Float64(l * h) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.25e-172)
		tmp = sqrt(((d / l) * (d / h)));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.25e-172], N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.25000000000000002e-172

   1. Initial program 74.6%

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

   3. Simplified 74.6%

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

      1. frac-times 74.6%

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

      2. associate-/r* 74.6%

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

   6. Applied egg-rr 74.6%

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

      1. expm1-log1p-u 39.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-udef 27.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]

   8. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 29.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(D \cdot \frac{M \cdot 0.5}{d}\right)}^{2}\right)\right)\right)\right)} \]

      2. expm1-log1p 61.6%

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

      3. associate-*r* 61.6%

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

      4. *-commutative 61.6%

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

      5. *-commutative 61.6%

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

      6. +-commutative 61.6%

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

      7. fma-def 61.6%

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

      8. *-commutative 61.6%

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

      9. associate-/r/ 62.7%

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

      10. *-commutative 62.7%

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

   10. Simplified 62.7%

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

   11. Taylor expanded in M around 0 32.9%

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

   if -2.25000000000000002e-172 < d

   1. Initial program 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in d around inf 34.3%

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

      1. *-commutative 34.3%

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

      2. associate-/r* 34.7%

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

   7. Simplified 34.7%

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

   8. Taylor expanded in d around 0 34.3%

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

      1. unpow1/2 34.3%

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

      2. rem-exp-log 32.9%

         \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

      3. exp-neg 32.9%

         \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

      4. exp-prod 33.4%

         \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

      5. distribute-lft-neg-out 33.4%

         \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

      6. distribute-rgt-neg-in 33.4%

         \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

      7. metadata-eval 33.4%

         \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

      8. exp-to-pow 34.9%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   10. Simplified 34.9%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.25 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ d \cdot {\left(\ell \cdot h\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (* d (pow (* l h) -0.5)))

C

double code(double d, double h, double l, double M, double D) {
	return d * pow((l * h), -0.5);
}

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 * ((l * h) ** (-0.5d0))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d * Math.pow((l * h), -0.5);
}

Python

def code(d, h, l, M, D):
	return d * math.pow((l * h), -0.5)

Julia

function code(d, h, l, M, D)
	return Float64(d * (Float64(l * h) ^ -0.5))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d * ((l * h) ^ -0.5);
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.9%

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

3. Simplified 65.9%

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

5. Taylor expanded in d around inf 23.3%

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

   1. *-commutative 23.3%

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

   2. associate-/r* 23.5%

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

7. Simplified 23.5%

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

8. Taylor expanded in d around 0 23.3%

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

   1. unpow1/2 23.3%

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

   2. rem-exp-log 22.4%

      \[\leadsto d \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

   3. exp-neg 22.4%

      \[\leadsto d \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

   4. exp-prod 22.7%

      \[\leadsto d \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

   5. distribute-lft-neg-out 22.7%

      \[\leadsto d \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

   6. distribute-rgt-neg-in 22.7%

      \[\leadsto d \cdot e^{\color{blue}{\log \left(h \cdot \ell\right) \cdot \left(-0.5\right)}} \]

   7. metadata-eval 22.7%

      \[\leadsto d \cdot e^{\log \left(h \cdot \ell\right) \cdot \color{blue}{-0.5}} \]

   8. exp-to-pow 23.6%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Simplified 23.6%

    \[\leadsto \color{blue}{d \cdot {\left(h \cdot \ell\right)}^{-0.5}} \]

11. Final simplification 23.6%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
12. Add Preprocessing

Alternative 21: 26.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 65.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 65.9%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\color{blue}{\frac{1}{\frac{2}{M}}} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   2. frac-times 65.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{1 \cdot D}{\frac{2}{M} \cdot d}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   3. *-un-lft-identity 65.4%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{\color{blue}{D}}{\frac{2}{M} \cdot d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   4. associate-*l/ 65.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{\color{blue}{\frac{2 \cdot d}{M}}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. *-un-lft-identity 65.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{\frac{2 \cdot d}{\color{blue}{1 \cdot M}}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   6. times-frac 65.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{\color{blue}{\frac{2}{1} \cdot \frac{d}{M}}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   7. metadata-eval 65.5%

      \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{\color{blue}{2} \cdot \frac{d}{M}}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

6. Applied egg-rr 65.5%

   \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - 0.5 \cdot \left({\color{blue}{\left(\frac{D}{2 \cdot \frac{d}{M}}\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

7. Taylor expanded in d around inf 23.3%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
8. Step-by-step derivation

   1. rem-exp-log 17.5%

      \[\leadsto \color{blue}{e^{\log d}} \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   2. unpow1/2 17.5%

      \[\leadsto e^{\log d} \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}} \]

   3. rem-exp-log 17.4%

      \[\leadsto e^{\log d} \cdot {\left(\frac{1}{\color{blue}{e^{\log \left(h \cdot \ell\right)}}}\right)}^{0.5} \]

   4. rec-exp 17.4%

      \[\leadsto e^{\log d} \cdot {\color{blue}{\left(e^{-\log \left(h \cdot \ell\right)}\right)}}^{0.5} \]

   5. exp-prod 17.7%

      \[\leadsto e^{\log d} \cdot \color{blue}{e^{\left(-\log \left(h \cdot \ell\right)\right) \cdot 0.5}} \]

   6. distribute-lft-neg-in 17.7%

      \[\leadsto e^{\log d} \cdot e^{\color{blue}{-\log \left(h \cdot \ell\right) \cdot 0.5}} \]

   7. *-commutative 17.7%

      \[\leadsto e^{\log d} \cdot e^{-\color{blue}{0.5 \cdot \log \left(h \cdot \ell\right)}} \]

   8. distribute-lft-neg-in 17.7%

      \[\leadsto e^{\log d} \cdot e^{\color{blue}{\left(-0.5\right) \cdot \log \left(h \cdot \ell\right)}} \]

   9. metadata-eval 17.7%

      \[\leadsto e^{\log d} \cdot e^{\color{blue}{-0.5} \cdot \log \left(h \cdot \ell\right)} \]

   10. log-pow 17.7%

       \[\leadsto e^{\log d} \cdot e^{\color{blue}{\log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

   11. exp-sum 17.6%

       \[\leadsto \color{blue}{e^{\log d + \log \left({\left(h \cdot \ell\right)}^{-0.5}\right)}} \]

   12. log-pow 17.6%

       \[\leadsto e^{\log d + \color{blue}{-0.5 \cdot \log \left(h \cdot \ell\right)}} \]

   13. metadata-eval 17.6%

       \[\leadsto e^{\log d + \color{blue}{\left(-0.5\right)} \cdot \log \left(h \cdot \ell\right)} \]

   14. distribute-lft-neg-in 17.6%

       \[\leadsto e^{\log d + \color{blue}{\left(-0.5 \cdot \log \left(h \cdot \ell\right)\right)}} \]

   15. log-pow 17.6%

       \[\leadsto e^{\log d + \left(-\color{blue}{\log \left({\left(h \cdot \ell\right)}^{0.5}\right)}\right)} \]

   16. unpow1/2 17.6%

       \[\leadsto e^{\log d + \left(-\log \color{blue}{\left(\sqrt{h \cdot \ell}\right)}\right)} \]

   17. sub-neg 17.6%

       \[\leadsto e^{\color{blue}{\log d - \log \left(\sqrt{h \cdot \ell}\right)}} \]

   18. log-div 18.5%

       \[\leadsto e^{\color{blue}{\log \left(\frac{d}{\sqrt{h \cdot \ell}}\right)}} \]

9. Simplified 23.6%

   \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

10. Final simplification 23.6%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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)))))