Henrywood and Agarwal, Equation (12)

Percentage Accurate: 65.8% → 83.7%

Time: 25.7s

Alternatives: 21

Speedup: 1.4×

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: 65.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\\ t_1 := \sqrt{-d}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{t\_1}{\sqrt{-h}} \cdot \left(\frac{t\_1}{\sqrt{-\ell}} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* h (* 0.125 (/ (pow (/ (* M D) d) 2.0) l)))))
        (t_1 (sqrt (- d))))
   (if (<= l -2e-310)
     (* (/ t_1 (sqrt (- h))) (* (/ t_1 (sqrt (- l))) t_0))
     (* (/ (sqrt d) (sqrt h)) (* t_0 (sqrt (/ d l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (h * (0.125 * (pow(((M * D) / d), 2.0) / l)));
	double t_1 = sqrt(-d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * t_0);
	} else {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (h * (0.125d0 * ((((m * d_1) / d) ** 2.0d0) / l)))
    t_1 = sqrt(-d)
    if (l <= (-2d-310)) then
        tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * t_0)
    else
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (h * (0.125 * (Math.pow(((M * D) / d), 2.0) / l)));
	double t_1 = Math.sqrt(-d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (t_1 / Math.sqrt(-h)) * ((t_1 / Math.sqrt(-l)) * t_0);
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (h * (0.125 * (math.pow(((M * D) / d), 2.0) / l)))
	t_1 = math.sqrt(-d)
	tmp = 0
	if l <= -2e-310:
		tmp = (t_1 / math.sqrt(-h)) * ((t_1 / math.sqrt(-l)) * t_0)
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * math.sqrt((d / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l))))
	t_1 = sqrt(Float64(-d))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(t_1 / sqrt(Float64(-h))) * Float64(Float64(t_1 / sqrt(Float64(-l))) * t_0));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (h * (0.125 * ((((M * D) / d) ^ 2.0) / l)));
	t_1 = sqrt(-d);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (t_1 / sqrt(-h)) * ((t_1 / sqrt(-l)) * t_0);
	else
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(t$95$1 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in h around -inf 43.3%

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

      1. associate-*r* 43.3%

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

      2. neg-mul-1 43.3%

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

      3. sub-neg 43.3%

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

      4. distribute-lft-in 43.3%

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

   7. Simplified 62.4%

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

      1. associate-*r/ 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. frac-2neg 63.2%

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

       2. sqrt-div 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. frac-2neg 79.2%

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

       2. sqrt-div 87.3%

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

   13. Applied egg-rr 87.3%

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

   if -1.999999999999994e-310 < l

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in h around -inf 45.2%

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

      1. associate-*r* 45.2%

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

      2. neg-mul-1 45.2%

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

      3. sub-neg 45.2%

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

      4. distribute-lft-in 45.2%

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

   7. Simplified 68.5%

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

      1. associate-*r/ 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. sqrt-div 79.9%

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

   11. Applied egg-rr 79.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-d}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{t\_0}{\sqrt{-h}} \cdot \left(\frac{t\_0}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (- d))))
   (if (<= h -5e-312)
     (*
      (/ t_0 (sqrt (- h)))
      (*
       (/ t_0 (sqrt (- l)))
       (- 1.0 (* h (* 0.125 (/ (pow (* M (/ D d)) 2.0) l))))))
     (*
      (/ (sqrt d) (sqrt h))
      (*
       (- 1.0 (* h (* 0.125 (/ (pow (/ (* M D) d) 2.0) l))))
       (sqrt (/ d l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(-d);
	double tmp;
	if (h <= -5e-312) {
		tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0 - (h * (0.125 * (pow((M * (D / d)), 2.0) / l)))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (pow(((M * D) / d), 2.0) / l)))) * sqrt((d / l)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-d)
    if (h <= (-5d-312)) then
        tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0d0 - (h * (0.125d0 * (((m * (d_1 / d)) ** 2.0d0) / l)))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 - (h * (0.125d0 * ((((m * d_1) / d) ** 2.0d0) / l)))) * sqrt((d / l)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(-d);
	double tmp;
	if (h <= -5e-312) {
		tmp = (t_0 / Math.sqrt(-h)) * ((t_0 / Math.sqrt(-l)) * (1.0 - (h * (0.125 * (Math.pow((M * (D / d)), 2.0) / l)))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 - (h * (0.125 * (Math.pow(((M * D) / d), 2.0) / l)))) * Math.sqrt((d / l)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(-d)
	tmp = 0
	if h <= -5e-312:
		tmp = (t_0 / math.sqrt(-h)) * ((t_0 / math.sqrt(-l)) * (1.0 - (h * (0.125 * (math.pow((M * (D / d)), 2.0) / l)))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 - (h * (0.125 * (math.pow(((M * D) / d), 2.0) / l)))) * math.sqrt((d / l)))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(-d))
	tmp = 0.0
	if (h <= -5e-312)
		tmp = Float64(Float64(t_0 / sqrt(Float64(-h))) * Float64(Float64(t_0 / sqrt(Float64(-l))) * Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(M * Float64(D / d)) ^ 2.0) / l))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))) * sqrt(Float64(d / l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(-d);
	tmp = 0.0;
	if (h <= -5e-312)
		tmp = (t_0 / sqrt(-h)) * ((t_0 / sqrt(-l)) * (1.0 - (h * (0.125 * (((M * (D / d)) ^ 2.0) / l)))));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * ((((M * D) / d) ^ 2.0) / l)))) * sqrt((d / l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[(-d)], $MachinePrecision]}, If[LessEqual[h, -5e-312], N[(N[(t$95$0 / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -5.0000000000022e-312

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in h around -inf 43.3%

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

      1. associate-*r* 43.3%

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

      2. neg-mul-1 43.3%

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

      3. sub-neg 43.3%

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

      4. distribute-lft-in 43.3%

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

   7. Simplified 62.4%

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

      1. frac-2neg 79.2%

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

      2. sqrt-div 87.3%

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

   9. Applied egg-rr 68.3%

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

       1. frac-2neg 63.2%

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

       2. sqrt-div 79.2%

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

   11. Applied egg-rr 85.8%

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

   if -5.0000000000022e-312 < h

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in h around -inf 45.2%

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

      1. associate-*r* 45.2%

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

      2. neg-mul-1 45.2%

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

      3. sub-neg 45.2%

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

      4. distribute-lft-in 45.2%

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

   7. Simplified 68.5%

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

      1. associate-*r/ 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. sqrt-div 79.9%

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

   11. Applied egg-rr 79.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_1 \cdot \frac{t\_1}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+238}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{t\_1}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* M (/ D d))))
   (if (<= l -2e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* t_0 (- 1.0 (* h (* 0.125 (* t_1 (/ t_1 l)))))))
     (if (<= l 8.2e+238)
       (*
        (/ (sqrt d) (sqrt h))
        (* t_0 (- 1.0 (* h (* 0.125 (/ (pow t_1 2.0) l))))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = M * (D / d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))));
	} else if (l <= 8.2e+238) {
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (pow(t_1, 2.0) / l)))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = m * (d_1 / d)
    if (l <= (-2d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0d0 - (h * (0.125d0 * (t_1 * (t_1 / l))))))
    else if (l <= 8.2d+238) then
        tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0d0 - (h * (0.125d0 * ((t_1 ** 2.0d0) / l)))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = M * (D / d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))));
	} else if (l <= 8.2e+238) {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (Math.pow(t_1, 2.0) / l)))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = M * (D / d)
	tmp = 0
	if l <= -2e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))))
	elif l <= 8.2e+238:
		tmp = (math.sqrt(d) / math.sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * (math.pow(t_1, 2.0) / l)))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_1 * Float64(t_1 / l)))))));
	elseif (l <= 8.2e+238)
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64((t_1 ^ 2.0) / l))))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = M * (D / d);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))));
	elseif (l <= 8.2e+238)
		tmp = (sqrt(d) / sqrt(h)) * (t_0 * (1.0 - (h * (0.125 * ((t_1 ^ 2.0) / l)))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.2e+238], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in h around -inf 43.3%

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

      1. associate-*r* 43.3%

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

      2. neg-mul-1 43.3%

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

      3. sub-neg 43.3%

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

      4. distribute-lft-in 43.3%

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

   7. Simplified 62.4%

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

      1. associate-*r/ 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. frac-2neg 63.2%

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

       2. sqrt-div 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. unpow2 79.2%

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

       2. *-un-lft-identity 79.2%

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

       3. times-frac 79.4%

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

       4. associate-/l* 78.6%

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

       5. associate-/l* 78.6%

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

   13. Applied egg-rr 78.6%

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

   if -1.999999999999994e-310 < l < 8.1999999999999998e238

   1. Initial program 68.9%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in h around -inf 46.9%

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

      1. associate-*r* 46.9%

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

      2. neg-mul-1 46.9%

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

      3. sub-neg 46.9%

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

      4. distribute-lft-in 46.9%

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

   7. Simplified 70.8%

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

      1. sqrt-div 82.2%

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

   9. Applied egg-rr 82.2%

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

   if 8.1999999999999998e238 < l

   1. Initial program 52.2%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in d around inf 50.2%

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

      1. expm1-log1p-u 47.1%

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

      2. expm1-undefine 47.1%

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

   7. Applied egg-rr 47.1%

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

      1. sub-neg 47.1%

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

      2. metadata-eval 47.1%

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

      3. +-commutative 47.1%

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

      4. log1p-undefine 47.1%

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

      5. rem-exp-log 50.2%

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

      6. +-commutative 50.2%

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

      7. fma-define 50.2%

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

   9. Simplified 50.2%

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

       1. sqrt-div 50.1%

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

       2. metadata-eval 50.1%

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

       3. un-div-inv 50.1%

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

       4. +-commutative 50.1%

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

       5. fma-undefine 50.1%

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

       6. associate-+l+ 50.1%

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

       7. metadata-eval 50.1%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 50.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 50.1%

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

   13. Simplified 50.1%

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

       1. *-commutative 50.1%

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

       2. sqrt-prod 85.3%

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

   15. Applied egg-rr 85.3%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+238}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(M \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (- 1.0 (* h (* 0.125 (/ (pow (/ (* M D) d) 2.0) l))))
          (sqrt (/ d l)))))
   (if (<= h -5e-312)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (* (/ (sqrt d) (sqrt h)) t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (h * (0.125 * (pow(((M * D) / d), 2.0) / l)))) * sqrt((d / l));
	double tmp;
	if (h <= -5e-312) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - (h * (0.125d0 * ((((m * d_1) / d) ** 2.0d0) / l)))) * sqrt((d / l))
    if (h <= (-5d-312)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else
        tmp = (sqrt(d) / sqrt(h)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = (1.0 - (h * (0.125 * (Math.pow(((M * D) / d), 2.0) / l)))) * Math.sqrt((d / l));
	double tmp;
	if (h <= -5e-312) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = (1.0 - (h * (0.125 * (math.pow(((M * D) / d), 2.0) / l)))) * math.sqrt((d / l))
	tmp = 0
	if h <= -5e-312:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))) * sqrt(Float64(d / l)))
	tmp = 0.0
	if (h <= -5e-312)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = (1.0 - (h * (0.125 * ((((M * D) / d) ^ 2.0) / l)))) * sqrt((d / l));
	tmp = 0.0;
	if (h <= -5e-312)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	else
		tmp = (sqrt(d) / sqrt(h)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -5e-312], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if h < -5.0000000000022e-312

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in h around -inf 43.3%

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

      1. associate-*r* 43.3%

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

      2. neg-mul-1 43.3%

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

      3. sub-neg 43.3%

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

      4. distribute-lft-in 43.3%

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

   7. Simplified 62.4%

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

      1. associate-*r/ 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. frac-2neg 63.2%

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

       2. sqrt-div 79.2%

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

   11. Applied egg-rr 79.2%

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

   if -5.0000000000022e-312 < h

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in h around -inf 45.2%

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

      1. associate-*r* 45.2%

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

      2. neg-mul-1 45.2%

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

      3. sub-neg 45.2%

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

      4. distribute-lft-in 45.2%

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

   7. Simplified 68.5%

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

      1. associate-*r/ 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. sqrt-div 79.9%

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

   11. Applied egg-rr 79.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \leq -5 \cdot 10^{-312}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(t\_0 \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_1 \cdot \frac{t\_1}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l))) (t_1 (* M (/ D d))))
   (if (<= l -2e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* t_0 (- 1.0 (* h (* 0.125 (* t_1 (/ t_1 l)))))))
     (*
      (/ (sqrt d) (sqrt h))
      (* (- 1.0 (* h (* 0.125 (/ (pow (/ (* M D) d) 2.0) l)))) t_0)))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l));
	double t_1 = M * (D / d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))));
	} else {
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * (pow(((M * D) / d), 2.0) / l)))) * t_0);
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((d / l))
    t_1 = m * (d_1 / d)
    if (l <= (-2d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0d0 - (h * (0.125d0 * (t_1 * (t_1 / l))))))
    else
        tmp = (sqrt(d) / sqrt(h)) * ((1.0d0 - (h * (0.125d0 * ((((m * d_1) / d) ** 2.0d0) / l)))) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = M * (D / d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * ((1.0 - (h * (0.125 * (Math.pow(((M * D) / d), 2.0) / l)))) * t_0);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l))
	t_1 = M * (D / d)
	tmp = 0
	if l <= -2e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * ((1.0 - (h * (0.125 * (math.pow(((M * D) / d), 2.0) / l)))) * t_0)
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(t_0 * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_1 * Float64(t_1 / l)))))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * Float64(Float64(1.0 - Float64(h * Float64(0.125 * Float64((Float64(Float64(M * D) / d) ^ 2.0) / l)))) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l));
	t_1 = M * (D / d);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (t_0 * (1.0 - (h * (0.125 * (t_1 * (t_1 / l))))));
	else
		tmp = (sqrt(d) / sqrt(h)) * ((1.0 - (h * (0.125 * ((((M * D) / d) ^ 2.0) / l)))) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 - N[(h * N[(0.125 * N[(t$95$1 * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(h * N[(0.125 * N[(N[Power[N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in h around -inf 43.3%

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

      1. associate-*r* 43.3%

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

      2. neg-mul-1 43.3%

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

      3. sub-neg 43.3%

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

      4. distribute-lft-in 43.3%

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

   7. Simplified 62.4%

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

      1. associate-*r/ 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. frac-2neg 63.2%

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

       2. sqrt-div 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. unpow2 79.2%

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

       2. *-un-lft-identity 79.2%

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

       3. times-frac 79.4%

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

       4. associate-/l* 78.6%

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

       5. associate-/l* 78.6%

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

   13. Applied egg-rr 78.6%

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

   if -1.999999999999994e-310 < l

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in h around -inf 45.2%

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

      1. associate-*r* 45.2%

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

      2. neg-mul-1 45.2%

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

      3. sub-neg 45.2%

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

      4. distribute-lft-in 45.2%

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

   7. Simplified 68.5%

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

      1. associate-*r/ 68.5%

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

   9. Applied egg-rr 68.5%

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

       1. sqrt-div 79.9%

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

   11. Applied egg-rr 79.9%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\left(1 - h \cdot \left(0.125 \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{2}}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \frac{D}{d}\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(t\_0 \cdot \frac{t\_0}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+184}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* M (/ D d))))
   (if (<= l -2e-310)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (sqrt (/ d l)) (- 1.0 (* h (* 0.125 (* t_0 (/ t_0 l)))))))
     (if (<= l 3.1e+184)
       (*
        (* d (pow (* l h) -0.5))
        (- 1.0 (* 0.5 (* (pow (* (/ D d) (/ M 2.0)) 2.0) (/ h l)))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D / d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= 3.1e+184) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * (pow(((D / d) * (M / 2.0)), 2.0) * (h / l))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m * (d_1 / d)
    if (l <= (-2d-310)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (h * (0.125d0 * (t_0 * (t_0 / l))))))
    else if (l <= 3.1d+184) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - (0.5d0 * ((((d_1 / d) * (m / 2.0d0)) ** 2.0d0) * (h / l))))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = M * (D / d);
	double tmp;
	if (l <= -2e-310) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / l))))));
	} else if (l <= 3.1e+184) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - (0.5 * (Math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l))));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = M * (D / d)
	tmp = 0
	if l <= -2e-310:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / l))))))
	elif l <= 3.1e+184:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - (0.5 * (math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l))))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(M * Float64(D / d))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(h * Float64(0.125 * Float64(t_0 * Float64(t_0 / l)))))));
	elseif (l <= 3.1e+184)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = M * (D / d);
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (h * (0.125 * (t_0 * (t_0 / l))))));
	elseif (l <= 3.1e+184)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - (0.5 * ((((D / d) * (M / 2.0)) ^ 2.0) * (h / l))));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(M * N[(D / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(h * N[(0.125 * N[(t$95$0 * N[(t$95$0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.1e+184], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in h around -inf 43.3%

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

      1. associate-*r* 43.3%

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

      2. neg-mul-1 43.3%

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

      3. sub-neg 43.3%

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

      4. distribute-lft-in 43.3%

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

   7. Simplified 62.4%

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

      1. associate-*r/ 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. frac-2neg 63.2%

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

       2. sqrt-div 79.2%

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

   11. Applied egg-rr 79.2%

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

       1. unpow2 79.2%

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

       2. *-un-lft-identity 79.2%

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

       3. times-frac 79.4%

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

       4. associate-/l* 78.6%

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

       5. associate-/l* 78.6%

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

   13. Applied egg-rr 78.6%

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

   if -1.999999999999994e-310 < l < 3.0999999999999998e184

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around 0 80.3%

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

      1. unpow-1 80.3%

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

      2. metadata-eval 80.3%

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

      3. pow-sqr 80.3%

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

      4. rem-sqrt-square 80.8%

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

      5. rem-square-sqrt 80.5%

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

      6. fabs-sqr 80.5%

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

      7. rem-square-sqrt 80.8%

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

   9. Simplified 80.8%

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

   if 3.0999999999999998e184 < l

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in d around inf 40.3%

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

      1. expm1-log1p-u 37.7%

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

      2. expm1-undefine 36.9%

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

   7. Applied egg-rr 36.9%

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

      1. sub-neg 36.9%

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

      2. metadata-eval 36.9%

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

      3. +-commutative 36.9%

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

      4. log1p-undefine 36.9%

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

      5. rem-exp-log 39.4%

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

      6. +-commutative 39.4%

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

      7. fma-define 39.4%

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

   9. Simplified 39.4%

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

       1. sqrt-div 39.4%

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

       2. metadata-eval 39.4%

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

       3. un-div-inv 39.4%

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

       4. +-commutative 39.4%

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

       5. fma-undefine 39.4%

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

       6. associate-+l+ 40.2%

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

       7. metadata-eval 40.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 40.2%

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

   13. Simplified 40.2%

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

       1. *-commutative 40.2%

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

       2. sqrt-prod 69.1%

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

   15. Applied egg-rr 69.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - h \cdot \left(0.125 \cdot \left(\left(M \cdot \frac{D}{d}\right) \cdot \frac{M \cdot \frac{D}{d}}{\ell}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+184}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+93}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-306}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (- 1.0 (* 0.5 (* (pow (* (/ D d) (/ M 2.0)) 2.0) (/ h l))))))
   (if (<= l -4.4e+93)
     (* (- d) (sqrt (/ (/ 1.0 l) h)))
     (if (<= l 2e-306)
       (* t_0 (sqrt (* (/ d l) (/ d h))))
       (if (<= l 2.7e+181)
         (* (* d (pow (* l h) -0.5)) t_0)
         (/ d (* (sqrt h) (sqrt l))))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * (pow(((D / d) * (M / 2.0)), 2.0) * (h / l)));
	double tmp;
	if (l <= -4.4e+93) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 2e-306) {
		tmp = t_0 * sqrt(((d / l) * (d / h)));
	} else if (l <= 2.7e+181) {
		tmp = (d * pow((l * h), -0.5)) * t_0;
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.5d0 * ((((d_1 / d) * (m / 2.0d0)) ** 2.0d0) * (h / l)))
    if (l <= (-4.4d+93)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= 2d-306) then
        tmp = t_0 * sqrt(((d / l) * (d / h)))
    else if (l <= 2.7d+181) then
        tmp = (d * ((l * h) ** (-0.5d0))) * t_0
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 1.0 - (0.5 * (Math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l)));
	double tmp;
	if (l <= -4.4e+93) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= 2e-306) {
		tmp = t_0 * Math.sqrt(((d / l) * (d / h)));
	} else if (l <= 2.7e+181) {
		tmp = (d * Math.pow((l * h), -0.5)) * t_0;
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 1.0 - (0.5 * (math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l)))
	tmp = 0
	if l <= -4.4e+93:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= 2e-306:
		tmp = t_0 * math.sqrt(((d / l) * (d / h)))
	elif l <= 2.7e+181:
		tmp = (d * math.pow((l * h), -0.5)) * t_0
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(h / l))))
	tmp = 0.0
	if (l <= -4.4e+93)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 2e-306)
		tmp = Float64(t_0 * sqrt(Float64(Float64(d / l) * Float64(d / h))));
	elseif (l <= 2.7e+181)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * t_0);
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 1.0 - (0.5 * ((((D / d) * (M / 2.0)) ^ 2.0) * (h / l)));
	tmp = 0.0;
	if (l <= -4.4e+93)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= 2e-306)
		tmp = t_0 * sqrt(((d / l) * (d / h)));
	elseif (l <= 2.7e+181)
		tmp = (d * ((l * h) ^ -0.5)) * t_0;
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.4e+93], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e-306], N[(t$95$0 * N[Sqrt[N[(N[(d / l), $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.7e+181], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.40000000000000042e93

   1. Initial program 51.0%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in d around inf 2.8%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 65.2%

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

      5. *-commutative 65.2%

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

      6. associate-*r* 65.2%

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

      7. *-commutative 65.2%

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

      8. associate-/r* 66.8%

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

      9. neg-mul-1 66.8%

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

   8. Simplified 66.8%

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

   if -4.40000000000000042e93 < l < 2.00000000000000006e-306

   1. Initial program 68.5%

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

   3. Simplified 67.3%

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

      1. pow1 67.3%

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

      2. sqrt-unprod 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. unpow1 57.0%

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

   8. Simplified 57.0%

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

   if 2.00000000000000006e-306 < l < 2.70000000000000007e181

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around 0 80.3%

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

      1. unpow-1 80.3%

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

      2. metadata-eval 80.3%

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

      3. pow-sqr 80.3%

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

      4. rem-sqrt-square 80.8%

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

      5. rem-square-sqrt 80.5%

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

      6. fabs-sqr 80.5%

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

      7. rem-square-sqrt 80.8%

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

   9. Simplified 80.8%

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

   if 2.70000000000000007e181 < l

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in d around inf 40.3%

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

      1. expm1-log1p-u 37.7%

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

      2. expm1-undefine 36.9%

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

   7. Applied egg-rr 36.9%

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

      1. sub-neg 36.9%

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

      2. metadata-eval 36.9%

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

      3. +-commutative 36.9%

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

      4. log1p-undefine 36.9%

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

      5. rem-exp-log 39.4%

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

      6. +-commutative 39.4%

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

      7. fma-define 39.4%

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

   9. Simplified 39.4%

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

       1. sqrt-div 39.4%

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

       2. metadata-eval 39.4%

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

       3. un-div-inv 39.4%

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

       4. +-commutative 39.4%

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

       5. fma-undefine 39.4%

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

       6. associate-+l+ 40.2%

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

       7. metadata-eval 40.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 40.2%

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

   13. Simplified 40.2%

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

       1. *-commutative 40.2%

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

       2. sqrt-prod 69.1%

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

   15. Applied egg-rr 69.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+93}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-306}:\\ \;\;\;\;\left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{d}{\ell} \cdot \frac{d}{h}}\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-172}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+182}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.5e-172)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -2e-310)
     (/ d (sqrt (+ -1.0 (fma h l 1.0))))
     (if (<= l 3.2e+182)
       (*
        (* d (pow (* l h) -0.5))
        (- 1.0 (* 0.5 (* (pow (* (/ D d) (/ M 2.0)) 2.0) (/ h l)))))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.5e-172) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / sqrt((-1.0 + fma(h, l, 1.0)));
	} else if (l <= 3.2e+182) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - (0.5 * (pow(((D / d) * (M / 2.0)), 2.0) * (h / l))));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.5e-172)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -2e-310)
		tmp = Float64(d / sqrt(Float64(-1.0 + fma(h, l, 1.0))));
	elseif (l <= 3.2e+182)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - Float64(0.5 * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(h / l)))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.5e-172], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Sqrt[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+182], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.49999999999999992e-172

   1. Initial program 55.5%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*r* 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-/r* 54.9%

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

      9. neg-mul-1 54.9%

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

   8. Simplified 54.9%

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

   if -1.49999999999999992e-172 < l < -1.999999999999994e-310

   1. Initial program 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in d around inf 33.5%

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

      1. expm1-log1p-u 33.5%

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

      2. expm1-undefine 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. sub-neg 47.9%

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

      2. metadata-eval 47.9%

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

      3. +-commutative 47.9%

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

      4. log1p-undefine 47.9%

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

      5. rem-exp-log 47.9%

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

      6. +-commutative 47.9%

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

      7. fma-define 47.9%

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

   9. Simplified 47.9%

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

       1. sqrt-div 47.9%

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

       2. metadata-eval 47.9%

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

       3. un-div-inv 47.9%

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

       4. +-commutative 47.9%

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

       5. fma-undefine 47.9%

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

       6. associate-+l+ 28.8%

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

       7. metadata-eval 28.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 28.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 28.8%

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

   13. Simplified 28.8%

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

       1. expm1-log1p-u 33.5%

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

       2. expm1-undefine 47.9%

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

   15. Applied egg-rr 47.9%

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

       1. sub-neg 47.9%

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

       2. metadata-eval 47.9%

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

       3. +-commutative 47.9%

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

       4. log1p-undefine 47.9%

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

       5. rem-exp-log 47.9%

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

       6. +-commutative 47.9%

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

       7. fma-define 47.9%

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

   17. Simplified 47.9%

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

   if -1.999999999999994e-310 < l < 3.1999999999999997e182

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around 0 80.3%

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

      1. unpow-1 80.3%

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

      2. metadata-eval 80.3%

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

      3. pow-sqr 80.3%

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

      4. rem-sqrt-square 80.8%

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

      5. rem-square-sqrt 80.5%

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

      6. fabs-sqr 80.5%

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

      7. rem-square-sqrt 80.8%

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

   9. Simplified 80.8%

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

   if 3.1999999999999997e182 < l

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in d around inf 40.3%

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

      1. expm1-log1p-u 37.7%

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

      2. expm1-undefine 36.9%

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

   7. Applied egg-rr 36.9%

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

      1. sub-neg 36.9%

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

      2. metadata-eval 36.9%

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

      3. +-commutative 36.9%

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

      4. log1p-undefine 36.9%

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

      5. rem-exp-log 39.4%

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

      6. +-commutative 39.4%

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

      7. fma-define 39.4%

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

   9. Simplified 39.4%

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

       1. sqrt-div 39.4%

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

       2. metadata-eval 39.4%

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

       3. un-div-inv 39.4%

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

       4. +-commutative 39.4%

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

       5. fma-undefine 39.4%

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

       6. associate-+l+ 40.2%

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

       7. metadata-eval 40.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 40.2%

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

   13. Simplified 40.2%

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

       1. *-commutative 40.2%

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

       2. sqrt-prod 69.1%

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

   15. Applied egg-rr 69.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5 \cdot 10^{-172}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+182}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-174}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-301}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-65}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(-0.125 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5.6e-174)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 1.95e-301)
     (/ d (sqrt (+ -1.0 (fma h l 1.0))))
     (if (<= l 2.45e-65)
       (* (/ d (sqrt (* l h))) (* -0.125 (* h (/ (pow (/ D (/ d M)) 2.0) l))))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.6e-174) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 1.95e-301) {
		tmp = d / sqrt((-1.0 + fma(h, l, 1.0)));
	} else if (l <= 2.45e-65) {
		tmp = (d / sqrt((l * h))) * (-0.125 * (h * (pow((D / (d / M)), 2.0) / l)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5.6e-174)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 1.95e-301)
		tmp = Float64(d / sqrt(Float64(-1.0 + fma(h, l, 1.0))));
	elseif (l <= 2.45e-65)
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(-0.125 * Float64(h * Float64((Float64(D / Float64(d / M)) ^ 2.0) / l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5.6e-174], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e-301], N[(d / N[Sqrt[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.45e-65], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.125 * N[(h * N[(N[Power[N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.59999999999999998e-174

   1. Initial program 55.5%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*r* 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-/r* 54.9%

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

      9. neg-mul-1 54.9%

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

   8. Simplified 54.9%

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

   if -5.59999999999999998e-174 < l < 1.9500000000000001e-301

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in d around inf 35.1%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-undefine 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. sub-neg 49.1%

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

      2. metadata-eval 49.1%

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

      3. +-commutative 49.1%

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

      4. log1p-undefine 49.1%

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

      5. rem-exp-log 49.1%

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

      6. +-commutative 49.1%

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

      7. fma-define 49.1%

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

   9. Simplified 49.1%

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

       1. sqrt-div 49.1%

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

       2. metadata-eval 49.1%

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

       3. un-div-inv 49.1%

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

       4. +-commutative 49.1%

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

       5. fma-undefine 49.1%

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

       6. associate-+l+ 30.5%

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

       7. metadata-eval 30.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 30.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 30.5%

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

   13. Simplified 30.5%

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

       1. expm1-log1p-u 35.1%

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

       2. expm1-undefine 49.1%

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

   15. Applied egg-rr 49.1%

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

       1. sub-neg 49.1%

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

       2. metadata-eval 49.1%

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

       3. +-commutative 49.1%

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

       4. log1p-undefine 49.1%

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

       5. rem-exp-log 49.1%

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

       6. +-commutative 49.1%

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

       7. fma-define 49.1%

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

   17. Simplified 49.1%

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

   if 1.9500000000000001e-301 < l < 2.44999999999999982e-65

   1. Initial program 70.2%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in M around inf 28.1%

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

      1. associate-*r* 30.4%

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

      2. times-frac 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-/l* 30.5%

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

      5. unpow2 30.5%

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

      6. unpow2 30.5%

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

      7. unpow2 30.5%

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

      8. times-frac 34.7%

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

      9. swap-sqr 43.2%

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

      10. unpow2 43.2%

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

   7. Simplified 43.2%

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

      1. pow1 43.2%

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

      2. associate-*r* 43.1%

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

      3. sqrt-div 62.4%

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

      4. sqrt-div 62.4%

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

      5. frac-times 62.4%

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

      6. add-sqr-sqrt 62.4%

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

      7. sqrt-prod 62.3%

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

   9. Applied egg-rr 62.3%

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

       1. unpow1 62.3%

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

       2. associate-*r/ 62.5%

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

       3. *-commutative 62.5%

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

       4. associate-/l* 62.5%

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

       5. *-commutative 62.5%

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

       6. associate-/r/ 62.5%

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

   11. Simplified 62.5%

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

   if 2.44999999999999982e-65 < l

   1. Initial program 64.5%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in d around inf 48.0%

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

      1. *-un-lft-identity 48.0%

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

      2. pow1/2 48.0%

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

      3. inv-pow 48.0%

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

      4. pow-pow 48.1%

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

      5. metadata-eval 48.1%

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

   7. Applied egg-rr 48.1%

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

      1. *-lft-identity 48.1%

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

   9. Simplified 48.1%

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

       1. *-commutative 48.1%

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

       2. unpow-prod-down 60.4%

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

   11. Applied egg-rr 60.4%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-174}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{-301}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{-65}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(-0.125 \cdot \left(h \cdot \frac{{\left(\frac{D}{\frac{d}{M}}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(t\_0 + -1\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+182}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (pow (* (/ D d) (/ M 2.0)) 2.0) (/ h l)))))
   (if (<= l -2e-310)
     (* (* d (sqrt (/ 1.0 (* l h)))) (+ t_0 -1.0))
     (if (<= l 3.2e+182)
       (* (* d (pow (* l h) -0.5)) (- 1.0 t_0))
       (/ d (* (sqrt h) (sqrt l)))))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (pow(((D / d) * (M / 2.0)), 2.0) * (h / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (l <= 3.2e+182) {
		tmp = (d * pow((l * h), -0.5)) * (1.0 - t_0);
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((((d_1 / d) * (m / 2.0d0)) ** 2.0d0) * (h / l))
    if (l <= (-2d-310)) then
        tmp = (d * sqrt((1.0d0 / (l * h)))) * (t_0 + (-1.0d0))
    else if (l <= 3.2d+182) then
        tmp = (d * ((l * h) ** (-0.5d0))) * (1.0d0 - t_0)
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = 0.5 * (Math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l));
	double tmp;
	if (l <= -2e-310) {
		tmp = (d * Math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	} else if (l <= 3.2e+182) {
		tmp = (d * Math.pow((l * h), -0.5)) * (1.0 - t_0);
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = 0.5 * (math.pow(((D / d) * (M / 2.0)), 2.0) * (h / l))
	tmp = 0
	if l <= -2e-310:
		tmp = (d * math.sqrt((1.0 / (l * h)))) * (t_0 + -1.0)
	elif l <= 3.2e+182:
		tmp = (d * math.pow((l * h), -0.5)) * (1.0 - t_0)
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = Float64(0.5 * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(h / l)))
	tmp = 0.0
	if (l <= -2e-310)
		tmp = Float64(Float64(d * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(t_0 + -1.0));
	elseif (l <= 3.2e+182)
		tmp = Float64(Float64(d * (Float64(l * h) ^ -0.5)) * Float64(1.0 - t_0));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = 0.5 * ((((D / d) * (M / 2.0)) ^ 2.0) * (h / l));
	tmp = 0.0;
	if (l <= -2e-310)
		tmp = (d * sqrt((1.0 / (l * h)))) * (t_0 + -1.0);
	elseif (l <= 3.2e+182)
		tmp = (d * ((l * h) ^ -0.5)) * (1.0 - t_0);
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(0.5 * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e-310], N[(N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+182], N[(N[(d * N[Power[N[(l * h), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.999999999999994e-310

   1. Initial program 62.2%

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

   3. Simplified 61.4%

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

      1. frac-2neg 63.2%

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

      2. sqrt-div 79.2%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in d around -inf 72.1%

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

   if -1.999999999999994e-310 < l < 3.1999999999999997e182

   1. Initial program 71.6%

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

   3. Simplified 71.6%

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

      1. frac-2neg 72.8%

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

      2. sqrt-div 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in d around 0 80.3%

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

      1. unpow-1 80.3%

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

      2. metadata-eval 80.3%

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

      3. pow-sqr 80.3%

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

      4. rem-sqrt-square 80.8%

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

      5. rem-square-sqrt 80.5%

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

      6. fabs-sqr 80.5%

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

      7. rem-square-sqrt 80.8%

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

   9. Simplified 80.8%

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

   if 3.1999999999999997e182 < l

   1. Initial program 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in d around inf 40.3%

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

      1. expm1-log1p-u 37.7%

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

      2. expm1-undefine 36.9%

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

   7. Applied egg-rr 36.9%

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

      1. sub-neg 36.9%

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

      2. metadata-eval 36.9%

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

      3. +-commutative 36.9%

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

      4. log1p-undefine 36.9%

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

      5. rem-exp-log 39.4%

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

      6. +-commutative 39.4%

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

      7. fma-define 39.4%

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

   9. Simplified 39.4%

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

       1. sqrt-div 39.4%

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

       2. metadata-eval 39.4%

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

       3. un-div-inv 39.4%

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

       4. +-commutative 39.4%

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

       5. fma-undefine 39.4%

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

       6. associate-+l+ 40.2%

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

       7. metadata-eval 40.2%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 40.2%

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

   13. Simplified 40.2%

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

       1. *-commutative 40.2%

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

       2. sqrt-prod 69.1%

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

   15. Applied egg-rr 69.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(d \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right) + -1\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+182}:\\ \;\;\;\;\left(d \cdot {\left(\ell \cdot h\right)}^{-0.5}\right) \cdot \left(1 - 0.5 \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.42 \cdot 10^{-162}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{1.5}}{\frac{d}{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.42e-162)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l 3.5e-301)
     (/ d (sqrt (+ -1.0 (fma h l 1.0))))
     (if (<= l 8.5e-254)
       (* (sqrt (/ d h)) (/ (pow (/ d l) 1.5) (/ d (- l))))
       (* d (* (pow l -0.5) (pow h -0.5)))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.42e-162) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= 3.5e-301) {
		tmp = d / sqrt((-1.0 + fma(h, l, 1.0)));
	} else if (l <= 8.5e-254) {
		tmp = sqrt((d / h)) * (pow((d / l), 1.5) / (d / -l));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.42e-162)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= 3.5e-301)
		tmp = Float64(d / sqrt(Float64(-1.0 + fma(h, l, 1.0))));
	elseif (l <= 8.5e-254)
		tmp = Float64(sqrt(Float64(d / h)) * Float64((Float64(d / l) ^ 1.5) / Float64(d / Float64(-l))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.42e-162], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.5e-301], N[(d / N[Sqrt[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e-254], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[(d / l), $MachinePrecision], 1.5], $MachinePrecision] / N[(d / (-l)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.41999999999999989e-162

   1. Initial program 55.5%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*r* 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-/r* 54.9%

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

      9. neg-mul-1 54.9%

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

   8. Simplified 54.9%

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

   if -1.41999999999999989e-162 < l < 3.49999999999999992e-301

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in d around inf 35.1%

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

      1. expm1-log1p-u 35.1%

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

      2. expm1-undefine 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. sub-neg 49.1%

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

      2. metadata-eval 49.1%

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

      3. +-commutative 49.1%

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

      4. log1p-undefine 49.1%

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

      5. rem-exp-log 49.1%

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

      6. +-commutative 49.1%

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

      7. fma-define 49.1%

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

   9. Simplified 49.1%

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

       1. sqrt-div 49.1%

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

       2. metadata-eval 49.1%

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

       3. un-div-inv 49.1%

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

       4. +-commutative 49.1%

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

       5. fma-undefine 49.1%

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

       6. associate-+l+ 30.5%

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

       7. metadata-eval 30.5%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 30.5%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 30.5%

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

   13. Simplified 30.5%

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

       1. expm1-log1p-u 35.1%

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

       2. expm1-undefine 49.1%

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

   15. Applied egg-rr 49.1%

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

       1. sub-neg 49.1%

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

       2. metadata-eval 49.1%

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

       3. +-commutative 49.1%

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

       4. log1p-undefine 49.1%

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

       5. rem-exp-log 49.1%

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

       6. +-commutative 49.1%

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

       7. fma-define 49.1%

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

   17. Simplified 49.1%

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

   if 3.49999999999999992e-301 < l < 8.49999999999999963e-254

   1. Initial program 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in l around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 17.3%

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

      4. mul-1-neg 17.3%

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

   7. Simplified 17.3%

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

      1. neg-sub0 17.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(0 - \sqrt{\frac{d}{\ell}}\right)} \]

      2. flip3-- 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{{0}^{3} - {\left(\sqrt{\frac{d}{\ell}}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)}} \]

      3. metadata-eval 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{0} - {\left(\sqrt{\frac{d}{\ell}}\right)}^{3}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      4. pow3 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{\ell}}}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      5. add-sqr-sqrt 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - \color{blue}{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      6. pow1 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - \color{blue}{{\left(\frac{d}{\ell}\right)}^{1}} \cdot \sqrt{\frac{d}{\ell}}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      7. pow1/2 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - {\left(\frac{d}{\ell}\right)}^{1} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      8. pow-prod-up 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(1 + 0.5\right)}}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      9. metadata-eval 71.7%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - {\left(\frac{d}{\ell}\right)}^{\color{blue}{1.5}}}{0 \cdot 0 + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      10. metadata-eval 71.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - {\left(\frac{d}{\ell}\right)}^{1.5}}{\color{blue}{0} + \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

      11. add-sqr-sqrt 71.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{0 - {\left(\frac{d}{\ell}\right)}^{1.5}}{0 + \left(\color{blue}{\frac{d}{\ell}} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

   9. Applied egg-rr 71.7%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{0 - {\left(\frac{d}{\ell}\right)}^{1.5}}{0 + \left(\frac{d}{\ell} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)}} \]
   10. Step-by-step derivation

       1. sub0-neg 71.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{\color{blue}{-{\left(\frac{d}{\ell}\right)}^{1.5}}}{0 + \left(\frac{d}{\ell} + 0 \cdot \sqrt{\frac{d}{\ell}}\right)} \]

       2. +-lft-identity 71.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{-{\left(\frac{d}{\ell}\right)}^{1.5}}{\color{blue}{\frac{d}{\ell} + 0 \cdot \sqrt{\frac{d}{\ell}}}} \]

       3. mul0-lft 71.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{-{\left(\frac{d}{\ell}\right)}^{1.5}}{\frac{d}{\ell} + \color{blue}{0}} \]

       4. +-rgt-identity 71.7%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \frac{-{\left(\frac{d}{\ell}\right)}^{1.5}}{\color{blue}{\frac{d}{\ell}}} \]

   11. Simplified 71.7%

       \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\frac{-{\left(\frac{d}{\ell}\right)}^{1.5}}{\frac{d}{\ell}}} \]

   if 8.49999999999999963e-254 < l

   1. Initial program 65.5%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in d around inf 42.4%

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

      1. *-un-lft-identity 42.4%

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

      2. pow1/2 42.4%

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

      3. inv-pow 42.4%

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

      4. pow-pow 42.9%

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

      5. metadata-eval 42.9%

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

   7. Applied egg-rr 42.9%

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

      1. *-lft-identity 42.9%

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

   9. Simplified 42.9%

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

       1. *-commutative 42.9%

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

       2. unpow-prod-down 53.1%

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

   11. Applied egg-rr 53.1%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.42 \cdot 10^{-162}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \frac{{\left(\frac{d}{\ell}\right)}^{1.5}}{\frac{d}{-\ell}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-174}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -3e-174)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -2e-310)
     (/ d (sqrt (+ -1.0 (fma h l 1.0))))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -3e-174) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / sqrt((-1.0 + fma(h, l, 1.0)));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -3e-174)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -2e-310)
		tmp = Float64(d / sqrt(Float64(-1.0 + fma(h, l, 1.0))));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -3e-174], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Sqrt[N[(-1.0 + N[(h * l + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.00000000000000021e-174

   1. Initial program 55.5%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*r* 54.0%

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

      7. *-commutative 54.0%

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

      8. associate-/r* 54.9%

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

      9. neg-mul-1 54.9%

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

   8. Simplified 54.9%

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

   if -3.00000000000000021e-174 < l < -1.999999999999994e-310

   1. Initial program 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in d around inf 33.5%

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

      1. expm1-log1p-u 33.5%

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

      2. expm1-undefine 47.9%

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

   7. Applied egg-rr 47.9%

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

      1. sub-neg 47.9%

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

      2. metadata-eval 47.9%

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

      3. +-commutative 47.9%

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

      4. log1p-undefine 47.9%

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

      5. rem-exp-log 47.9%

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

      6. +-commutative 47.9%

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

      7. fma-define 47.9%

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

   9. Simplified 47.9%

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

       1. sqrt-div 47.9%

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

       2. metadata-eval 47.9%

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

       3. un-div-inv 47.9%

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

       4. +-commutative 47.9%

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

       5. fma-undefine 47.9%

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

       6. associate-+l+ 28.8%

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

       7. metadata-eval 28.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 28.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 28.8%

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

   13. Simplified 28.8%

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

       1. expm1-log1p-u 33.5%

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

       2. expm1-undefine 47.9%

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

   15. Applied egg-rr 47.9%

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

       1. sub-neg 47.9%

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

       2. metadata-eval 47.9%

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

       3. +-commutative 47.9%

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

       4. log1p-undefine 47.9%

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

       5. rem-exp-log 47.9%

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

       6. +-commutative 47.9%

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

       7. fma-define 47.9%

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

   17. Simplified 47.9%

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

   if -1.999999999999994e-310 < l

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. pow1/2 40.6%

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

      3. inv-pow 40.6%

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

      4. pow-pow 41.0%

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

      5. metadata-eval 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. *-lft-identity 41.0%

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

   9. Simplified 41.0%

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

       1. *-commutative 41.0%

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

       2. unpow-prod-down 50.6%

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

   11. Applied egg-rr 50.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-174}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{-185}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -8.2e-185)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -2e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (* d (* (pow l -0.5) (pow h -0.5))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.2e-185) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) * pow(h, -0.5));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -8.2e-185) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) * Math.pow(h, -0.5));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -8.2e-185)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -2e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) * (h ^ -0.5)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -8.2e-185], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] * N[Power[h, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.2e-185

   1. Initial program 57.0%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. associate-*r* 53.3%

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

      7. *-commutative 53.3%

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

      8. associate-/r* 54.2%

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

      9. neg-mul-1 54.2%

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

   8. Simplified 54.2%

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

   if -8.2e-185 < l < -1.999999999999994e-310

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in d around inf 36.0%

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

      1. expm1-log1p-u 36.0%

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

      2. expm1-undefine 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. sub-neg 48.8%

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

      2. metadata-eval 48.8%

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

      3. +-commutative 48.8%

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

      4. log1p-undefine 48.8%

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

      5. rem-exp-log 48.8%

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

      6. +-commutative 48.8%

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

      7. fma-define 48.8%

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

   9. Simplified 48.8%

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

       1. sqrt-div 48.8%

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

       2. metadata-eval 48.8%

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

       3. un-div-inv 48.8%

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

       4. +-commutative 48.8%

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

       5. fma-undefine 48.8%

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

       6. associate-+l+ 30.8%

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

       7. metadata-eval 30.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 30.8%

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

   13. Simplified 30.8%

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

       1. add-cbrt-cube 38.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 38.5%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 38.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 38.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 38.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 38.5%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 38.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   15. Applied egg-rr 38.5%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   16. Step-by-step derivation

       1. unpow1/3 38.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   17. Simplified 38.5%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. pow1/2 40.6%

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

      3. inv-pow 40.6%

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

      4. pow-pow 41.0%

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

      5. metadata-eval 41.0%

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

   7. Applied egg-rr 41.0%

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

      1. *-lft-identity 41.0%

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

   9. Simplified 41.0%

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

       1. *-commutative 41.0%

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

       2. unpow-prod-down 50.6%

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

   11. Applied egg-rr 50.6%

       \[\leadsto d \cdot \color{blue}{\left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{-185}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \left({\ell}^{-0.5} \cdot {h}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-180}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.7e-180)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -2e-310)
     (/ d (cbrt (pow (* l h) 1.5)))
     (/ d (* (sqrt h) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-180) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-180) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.7e-180)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -2e-310)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.7e-180], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.69999999999999991e-180

   1. Initial program 57.0%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. associate-*r* 53.3%

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

      7. *-commutative 53.3%

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

      8. associate-/r* 54.2%

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

      9. neg-mul-1 54.2%

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

   8. Simplified 54.2%

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

   if -1.69999999999999991e-180 < l < -1.999999999999994e-310

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in d around inf 36.0%

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

      1. expm1-log1p-u 36.0%

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

      2. expm1-undefine 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. sub-neg 48.8%

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

      2. metadata-eval 48.8%

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

      3. +-commutative 48.8%

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

      4. log1p-undefine 48.8%

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

      5. rem-exp-log 48.8%

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

      6. +-commutative 48.8%

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

      7. fma-define 48.8%

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

   9. Simplified 48.8%

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

       1. sqrt-div 48.8%

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

       2. metadata-eval 48.8%

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

       3. un-div-inv 48.8%

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

       4. +-commutative 48.8%

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

       5. fma-undefine 48.8%

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

       6. associate-+l+ 30.8%

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

       7. metadata-eval 30.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 30.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 30.8%

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

   13. Simplified 30.8%

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

       1. add-cbrt-cube 38.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 38.5%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 38.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 38.5%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 38.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. pow-prod-up 38.5%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

       7. metadata-eval 38.5%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   15. Applied egg-rr 38.5%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   16. Step-by-step derivation

       1. unpow1/3 38.5%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   17. Simplified 38.5%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if -1.999999999999994e-310 < l

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-undefine 24.8%

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

   7. Applied egg-rr 24.8%

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

      1. sub-neg 24.8%

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

      2. metadata-eval 24.8%

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

      3. +-commutative 24.8%

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

      4. log1p-undefine 24.8%

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

      5. rem-exp-log 26.0%

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

      6. +-commutative 26.0%

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

      7. fma-define 26.0%

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

   9. Simplified 26.0%

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

       1. sqrt-div 26.0%

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

       2. metadata-eval 26.0%

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

       3. un-div-inv 26.0%

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

       4. +-commutative 26.0%

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

       5. fma-undefine 26.0%

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

       6. associate-+l+ 41.0%

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

       7. metadata-eval 41.0%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 41.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 41.0%

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

   13. Simplified 41.0%

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

       1. *-commutative 41.0%

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

       2. sqrt-prod 50.6%

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

   15. Applied egg-rr 50.6%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-180}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-185}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.7e-185)
   (* (- d) (sqrt (/ (/ 1.0 l) h)))
   (if (<= l -2e-310)
     (* d (sqrt (/ 1.0 (* l h))))
     (/ d (* (sqrt h) (sqrt l))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-185) {
		tmp = -d * sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d * sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (sqrt(h) * sqrt(l));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.7d-185)) then
        tmp = -d * sqrt(((1.0d0 / l) / h))
    else if (l <= (-2d-310)) then
        tmp = d * sqrt((1.0d0 / (l * h)))
    else
        tmp = d / (sqrt(h) * sqrt(l))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.7e-185) {
		tmp = -d * Math.sqrt(((1.0 / l) / h));
	} else if (l <= -2e-310) {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	} else {
		tmp = d / (Math.sqrt(h) * Math.sqrt(l));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.7e-185:
		tmp = -d * math.sqrt(((1.0 / l) / h))
	elif l <= -2e-310:
		tmp = d * math.sqrt((1.0 / (l * h)))
	else:
		tmp = d / (math.sqrt(h) * math.sqrt(l))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.7e-185)
		tmp = Float64(Float64(-d) * sqrt(Float64(Float64(1.0 / l) / h)));
	elseif (l <= -2e-310)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	else
		tmp = Float64(d / Float64(sqrt(h) * sqrt(l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.7e-185)
		tmp = -d * sqrt(((1.0 / l) / h));
	elseif (l <= -2e-310)
		tmp = d * sqrt((1.0 / (l * h)));
	else
		tmp = d / (sqrt(h) * sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.7e-185], N[((-d) * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -2e-310], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d / N[(N[Sqrt[h], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.6999999999999999e-185

   1. Initial program 57.0%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. associate-*r* 53.3%

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

      7. *-commutative 53.3%

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

      8. associate-/r* 54.2%

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

      9. neg-mul-1 54.2%

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

   8. Simplified 54.2%

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

   if -1.6999999999999999e-185 < l < -1.999999999999994e-310

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in d around inf 36.0%

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

   if -1.999999999999994e-310 < l

   1. Initial program 66.8%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in d around inf 40.6%

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

      1. expm1-log1p-u 39.4%

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

      2. expm1-undefine 24.8%

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

   7. Applied egg-rr 24.8%

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

      1. sub-neg 24.8%

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

      2. metadata-eval 24.8%

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

      3. +-commutative 24.8%

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

      4. log1p-undefine 24.8%

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

      5. rem-exp-log 26.0%

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

      6. +-commutative 26.0%

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

      7. fma-define 26.0%

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

   9. Simplified 26.0%

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

       1. sqrt-div 26.0%

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

       2. metadata-eval 26.0%

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

       3. un-div-inv 26.0%

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

       4. +-commutative 26.0%

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

       5. fma-undefine 26.0%

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

       6. associate-+l+ 41.0%

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

       7. metadata-eval 41.0%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 41.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 41.0%

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

   13. Simplified 41.0%

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

       1. *-commutative 41.0%

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

       2. sqrt-prod 50.6%

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

   15. Applied egg-rr 50.6%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{-185}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-310}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (/ (/ 1.0 l) h))))
   (if (<= l -3.1e-183) (* (- d) t_0) (* d t_0))))

C

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -3.1e-183) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	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 = sqrt(((1.0d0 / l) / h))
    if (l <= (-3.1d-183)) then
        tmp = -d * t_0
    else
        tmp = d * t_0
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt(((1.0 / l) / h));
	double tmp;
	if (l <= -3.1e-183) {
		tmp = -d * t_0;
	} else {
		tmp = d * t_0;
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	t_0 = math.sqrt(((1.0 / l) / h))
	tmp = 0
	if l <= -3.1e-183:
		tmp = -d * t_0
	else:
		tmp = d * t_0
	return tmp

Julia

function code(d, h, l, M, D)
	t_0 = sqrt(Float64(Float64(1.0 / l) / h))
	tmp = 0.0
	if (l <= -3.1e-183)
		tmp = Float64(Float64(-d) * t_0);
	else
		tmp = Float64(d * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt(((1.0 / l) / h));
	tmp = 0.0;
	if (l <= -3.1e-183)
		tmp = -d * t_0;
	else
		tmp = d * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.1e-183], N[((-d) * t$95$0), $MachinePrecision], N[(d * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.1e-183

   1. Initial program 57.0%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in d around inf 3.9%

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

   6. Taylor expanded in h around -inf 0.0%

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

      1. associate-*l* 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 53.3%

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

      5. *-commutative 53.3%

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

      6. associate-*r* 53.3%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-1 \cdot d\right)} \]

      7. *-commutative 53.3%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \cdot \left(-1 \cdot d\right) \]

      8. associate-/r* 54.2%

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \cdot \left(-1 \cdot d\right) \]

      9. neg-mul-1 54.2%

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \color{blue}{\left(-d\right)} \]

   8. Simplified 54.2%

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)} \]

   if -3.1e-183 < l

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 39.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]

   7. Applied egg-rr 39.6%

      \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 39.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/r* 39.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Simplified 39.9%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-183}:\\ \;\;\;\;\left(-d\right) \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-185}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -1.55e-185) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ (/ 1.0 l) h)))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.55e-185) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-1.55d-185)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -1.55e-185) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -1.55e-185:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -1.55e-185)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -1.55e-185)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -1.55e-185], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.5499999999999998e-185

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 3.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 3.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

      2. expm1-undefine 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

   7. Applied egg-rr 7.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]
   8. Step-by-step derivation

      1. sub-neg 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)}}} \]

      2. metadata-eval 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}}} \]

      3. +-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}}}} \]

      4. log1p-undefine 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}}} \]

      5. rem-exp-log 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(1 + h \cdot \ell\right)}}} \]

      6. +-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(h \cdot \ell + 1\right)}}} \]

      7. fma-define 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}}} \]

   9. Simplified 7.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]
   10. Step-by-step derivation

       1. sqrt-div 7.0%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

       2. metadata-eval 7.0%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}} \]

       3. un-div-inv 7.0%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

       4. +-commutative 7.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{fma}\left(h, \ell, 1\right) + -1}}} \]

       5. fma-undefine 7.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(h \cdot \ell + 1\right)} + -1}} \]

       6. associate-+l+ 2.8%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell + \left(1 + -1\right)}}} \]

       7. metadata-eval 2.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 2.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 2.8%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   13. Simplified 2.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. frac-times 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. sqrt-div 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. add-sqr-sqrt 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       7. sqr-neg 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \]

       8. sqrt-unprod 0.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \]

       9. add-sqr-sqrt 3.8%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

       10. distribute-rgt-neg-out 3.8%

           \[\leadsto \color{blue}{-\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. sqrt-div 0.0%

           \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-div 0.0%

           \[\leadsto -\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

       13. frac-times 0.0%

           \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       14. add-sqr-sqrt 0.0%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       15. sqrt-prod 53.4%

           \[\leadsto -\frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       16. neg-sub0 53.4%

           \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{h \cdot \ell}}} \]

   15. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{h \cdot \ell}}} \]
   16. Step-by-step derivation

       1. neg-sub0 53.4%

          \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

       2. distribute-frac-neg2 53.4%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   17. Simplified 53.4%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -1.5499999999999998e-185 < l

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 39.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]

   7. Applied egg-rr 39.6%

      \[\leadsto d \cdot \sqrt{\color{blue}{1 \cdot \frac{1}{h \cdot \ell}}} \]
   8. Step-by-step derivation

      1. *-lft-identity 39.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}} \]

      2. *-commutative 39.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      3. associate-/r* 39.9%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   9. Simplified 39.9%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-185}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5.2e-185) (/ d (- (sqrt (* l h)))) (* d (sqrt (/ 1.0 (* l h))))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.2e-185) {
		tmp = d / -sqrt((l * h));
	} else {
		tmp = d * sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5.2d-185)) then
        tmp = d / -sqrt((l * h))
    else
        tmp = d * sqrt((1.0d0 / (l * h)))
    end if
    code = tmp
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5.2e-185) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.sqrt((1.0 / (l * h)));
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -5.2e-185:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.sqrt((1.0 / (l * h)))
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5.2e-185)
		tmp = Float64(d / Float64(-sqrt(Float64(l * h))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(l * h))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5.2e-185)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * sqrt((1.0 / (l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5.2e-185], N[(d / (-N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.1999999999999997e-185

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 3.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 3.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

      2. expm1-undefine 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

   7. Applied egg-rr 7.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]
   8. Step-by-step derivation

      1. sub-neg 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)}}} \]

      2. metadata-eval 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}}} \]

      3. +-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}}}} \]

      4. log1p-undefine 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}}} \]

      5. rem-exp-log 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(1 + h \cdot \ell\right)}}} \]

      6. +-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(h \cdot \ell + 1\right)}}} \]

      7. fma-define 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}}} \]

   9. Simplified 7.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]
   10. Step-by-step derivation

       1. sqrt-div 7.0%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

       2. metadata-eval 7.0%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}} \]

       3. un-div-inv 7.0%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

       4. +-commutative 7.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{fma}\left(h, \ell, 1\right) + -1}}} \]

       5. fma-undefine 7.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(h \cdot \ell + 1\right)} + -1}} \]

       6. associate-+l+ 2.8%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell + \left(1 + -1\right)}}} \]

       7. metadata-eval 2.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 2.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 2.8%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   13. Simplified 2.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. frac-times 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. sqrt-div 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. add-sqr-sqrt 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       7. sqr-neg 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \]

       8. sqrt-unprod 0.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \]

       9. add-sqr-sqrt 3.8%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

       10. distribute-rgt-neg-out 3.8%

           \[\leadsto \color{blue}{-\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. sqrt-div 0.0%

           \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-div 0.0%

           \[\leadsto -\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

       13. frac-times 0.0%

           \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       14. add-sqr-sqrt 0.0%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       15. sqrt-prod 53.4%

           \[\leadsto -\frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       16. neg-sub0 53.4%

           \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{h \cdot \ell}}} \]

   15. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{h \cdot \ell}}} \]
   16. Step-by-step derivation

       1. neg-sub0 53.4%

          \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

       2. distribute-frac-neg2 53.4%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   17. Simplified 53.4%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -5.1999999999999997e-185 < l

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 39.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-185}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{\ell \cdot h}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-185}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (if (<= l -2.35e-185) (/ d (- (sqrt (* l h)))) (* d (pow (* l h) -0.5))))

C

double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -2.35e-185) {
		tmp = d / -sqrt((l * 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 (l <= (-2.35d-185)) then
        tmp = d / -sqrt((l * 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 (l <= -2.35e-185) {
		tmp = d / -Math.sqrt((l * h));
	} else {
		tmp = d * Math.pow((l * h), -0.5);
	}
	return tmp;
}

Python

def code(d, h, l, M, D):
	tmp = 0
	if l <= -2.35e-185:
		tmp = d / -math.sqrt((l * h))
	else:
		tmp = d * math.pow((l * h), -0.5)
	return tmp

Julia

function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -2.35e-185)
		tmp = Float64(d / Float64(-sqrt(Float64(l * 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 (l <= -2.35e-185)
		tmp = d / -sqrt((l * h));
	else
		tmp = d * ((l * h) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[d_, h_, l_, M_, D_] := If[LessEqual[l, -2.35e-185], N[(d / (-N[Sqrt[N[(l * 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 l < -2.3500000000000001e-185

   1. Initial program 57.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 3.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 3.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

      2. expm1-undefine 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

   7. Applied egg-rr 7.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]
   8. Step-by-step derivation

      1. sub-neg 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)}}} \]

      2. metadata-eval 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}}} \]

      3. +-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}}}} \]

      4. log1p-undefine 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}}} \]

      5. rem-exp-log 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(1 + h \cdot \ell\right)}}} \]

      6. +-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(h \cdot \ell + 1\right)}}} \]

      7. fma-define 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}}} \]

   9. Simplified 7.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]
   10. Step-by-step derivation

       1. sqrt-div 7.0%

          \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

       2. metadata-eval 7.0%

          \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}} \]

       3. un-div-inv 7.0%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

       4. +-commutative 7.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{fma}\left(h, \ell, 1\right) + -1}}} \]

       5. fma-undefine 7.0%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(h \cdot \ell + 1\right)} + -1}} \]

       6. associate-+l+ 2.8%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell + \left(1 + -1\right)}}} \]

       7. metadata-eval 2.8%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

   11. Applied egg-rr 2.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
   12. Step-by-step derivation

       1. +-rgt-identity 2.8%

          \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

   13. Simplified 2.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. frac-times 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. sqrt-div 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       6. add-sqr-sqrt 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{\ell}}}} \]

       7. sqr-neg 37.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right) \cdot \left(-\sqrt{\frac{d}{\ell}}\right)}} \]

       8. sqrt-unprod 0.6%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(\sqrt{-\sqrt{\frac{d}{\ell}}} \cdot \sqrt{-\sqrt{\frac{d}{\ell}}}\right)} \]

       9. add-sqr-sqrt 3.8%

          \[\leadsto \sqrt{\frac{d}{h}} \cdot \color{blue}{\left(-\sqrt{\frac{d}{\ell}}\right)} \]

       10. distribute-rgt-neg-out 3.8%

           \[\leadsto \color{blue}{-\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

       11. sqrt-div 0.0%

           \[\leadsto -\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \sqrt{\frac{d}{\ell}} \]

       12. sqrt-div 0.0%

           \[\leadsto -\frac{\sqrt{d}}{\sqrt{h}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \]

       13. frac-times 0.0%

           \[\leadsto -\color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       14. add-sqr-sqrt 0.0%

           \[\leadsto -\frac{\color{blue}{d}}{\sqrt{h} \cdot \sqrt{\ell}} \]

       15. sqrt-prod 53.4%

           \[\leadsto -\frac{d}{\color{blue}{\sqrt{h \cdot \ell}}} \]

       16. neg-sub0 53.4%

           \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{h \cdot \ell}}} \]

   15. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{0 - \frac{d}{\sqrt{h \cdot \ell}}} \]
   16. Step-by-step derivation

       1. neg-sub0 53.4%

          \[\leadsto \color{blue}{-\frac{d}{\sqrt{h \cdot \ell}}} \]

       2. distribute-frac-neg2 53.4%

          \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   17. Simplified 53.4%

       \[\leadsto \color{blue}{\frac{d}{-\sqrt{h \cdot \ell}}} \]

   if -2.3500000000000001e-185 < l

   1. Initial program 68.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in d around inf 39.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 39.6%

         \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. pow1/2 39.6%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

      3. inv-pow 39.6%

         \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

      4. pow-pow 38.7%

         \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

      5. metadata-eval 38.7%

         \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

   7. Applied egg-rr 38.7%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. *-lft-identity 38.7%

         \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

   9. Simplified 38.7%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.35 \cdot 10^{-185}:\\ \;\;\;\;\frac{d}{-\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot {\left(\ell \cdot h\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 26.8% 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 64.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 63.8%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 27.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 27.4%

      \[\leadsto d \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. pow1/2 27.4%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{h \cdot \ell}\right)}^{0.5}}\right) \]

   3. inv-pow 27.4%

      \[\leadsto d \cdot \left(1 \cdot {\color{blue}{\left({\left(h \cdot \ell\right)}^{-1}\right)}}^{0.5}\right) \]

   4. pow-pow 26.5%

      \[\leadsto d \cdot \left(1 \cdot \color{blue}{{\left(h \cdot \ell\right)}^{\left(-1 \cdot 0.5\right)}}\right) \]

   5. metadata-eval 26.5%

      \[\leadsto d \cdot \left(1 \cdot {\left(h \cdot \ell\right)}^{\color{blue}{-0.5}}\right) \]

7. Applied egg-rr 26.5%

   \[\leadsto d \cdot \color{blue}{\left(1 \cdot {\left(h \cdot \ell\right)}^{-0.5}\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 26.5%

      \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

9. Simplified 26.5%

   \[\leadsto d \cdot \color{blue}{{\left(h \cdot \ell\right)}^{-0.5}} \]

10. Final simplification 26.5%

    \[\leadsto d \cdot {\left(\ell \cdot h\right)}^{-0.5} \]
11. Add Preprocessing

Alternative 21: 26.8% 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 64.6%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 63.8%

   \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left({\left(0.5 \cdot \left(M \cdot \frac{D}{d}\right)\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in d around inf 27.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 26.8%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(h \cdot \ell\right)\right)}}} \]

   2. expm1-undefine 22.3%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]

7. Applied egg-rr 22.3%

   \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} - 1}}} \]
8. Step-by-step derivation

   1. sub-neg 22.3%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \left(-1\right)}}} \]

   2. metadata-eval 22.3%

      \[\leadsto d \cdot \sqrt{\frac{1}{e^{\mathsf{log1p}\left(h \cdot \ell\right)} + \color{blue}{-1}}} \]

   3. +-commutative 22.3%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + e^{\mathsf{log1p}\left(h \cdot \ell\right)}}}} \]

   4. log1p-undefine 22.3%

      \[\leadsto d \cdot \sqrt{\frac{1}{-1 + e^{\color{blue}{\log \left(1 + h \cdot \ell\right)}}}} \]

   5. rem-exp-log 23.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(1 + h \cdot \ell\right)}}} \]

   6. +-commutative 23.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\left(h \cdot \ell + 1\right)}}} \]

   7. fma-define 23.0%

      \[\leadsto d \cdot \sqrt{\frac{1}{-1 + \color{blue}{\mathsf{fma}\left(h, \ell, 1\right)}}} \]

9. Simplified 23.0%

   \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]
10. Step-by-step derivation

    1. sqrt-div 22.9%

       \[\leadsto d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

    2. metadata-eval 22.9%

       \[\leadsto d \cdot \frac{\color{blue}{1}}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}} \]

    3. un-div-inv 22.9%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{-1 + \mathsf{fma}\left(h, \ell, 1\right)}}} \]

    4. +-commutative 22.9%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{\mathsf{fma}\left(h, \ell, 1\right) + -1}}} \]

    5. fma-undefine 22.9%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{\left(h \cdot \ell + 1\right)} + -1}} \]

    6. associate-+l+ 26.5%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell + \left(1 + -1\right)}}} \]

    7. metadata-eval 26.5%

       \[\leadsto \frac{d}{\sqrt{h \cdot \ell + \color{blue}{0}}} \]

11. Applied egg-rr 26.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell + 0}}} \]
12. Step-by-step derivation

    1. +-rgt-identity 26.5%

       \[\leadsto \frac{d}{\sqrt{\color{blue}{h \cdot \ell}}} \]

13. Simplified 26.5%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

14. Final simplification 26.5%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))))