Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 0.6

Time: 29.2s

Precision: binary64

Cost: 14025

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-53} \lor \neg \left(k \leq 2.9 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot \tan k\right) \cdot \sin k} \cdot \ell}{t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{-k}{\ell}}{\frac{-1}{t \cdot \frac{k}{\frac{\ell}{k}}}}}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -5e-53) (not (<= k 2.9e-103)))
   (* (/ (* (/ 2.0 (* (* k (tan k)) (sin k))) l) t) (/ l k))
   (* 2.0 (/ 1.0 (/ (* k (/ (- k) l)) (/ -1.0 (* t (/ k (/ l k)))))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) - 1.0));
}

↓

double code(double t, double l, double k) {
	double tmp;
	if ((k <= -5e-53) || !(k <= 2.9e-103)) {
		tmp = (((2.0 / ((k * tan(k)) * sin(k))) * l) / t) * (l / k);
	} else {
		tmp = 2.0 * (1.0 / ((k * (-k / l)) / (-1.0 / (t * (k / (l / k))))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((k <= (-5d-53)) .or. (.not. (k <= 2.9d-103))) then
        tmp = (((2.0d0 / ((k * tan(k)) * sin(k))) * l) / t) * (l / k)
    else
        tmp = 2.0d0 * (1.0d0 / ((k * (-k / l)) / ((-1.0d0) / (t * (k / (l / k))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) - 1.0));
}

↓

public static double code(double t, double l, double k) {
	double tmp;
	if ((k <= -5e-53) || !(k <= 2.9e-103)) {
		tmp = (((2.0 / ((k * Math.tan(k)) * Math.sin(k))) * l) / t) * (l / k);
	} else {
		tmp = 2.0 * (1.0 / ((k * (-k / l)) / (-1.0 / (t * (k / (l / k))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) - 1.0))

↓

def code(t, l, k):
	tmp = 0
	if (k <= -5e-53) or not (k <= 2.9e-103):
		tmp = (((2.0 / ((k * math.tan(k)) * math.sin(k))) * l) / t) * (l / k)
	else:
		tmp = 2.0 * (1.0 / ((k * (-k / l)) / (-1.0 / (t * (k / (l / k))))))
	return tmp

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) - 1.0)))
end

↓

function code(t, l, k)
	tmp = 0.0
	if ((k <= -5e-53) || !(k <= 2.9e-103))
		tmp = Float64(Float64(Float64(Float64(2.0 / Float64(Float64(k * tan(k)) * sin(k))) * l) / t) * Float64(l / k));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64(Float64(k * Float64(Float64(-k) / l)) / Float64(-1.0 / Float64(t * Float64(k / Float64(l / k)))))));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

↓

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((k <= -5e-53) || ~((k <= 2.9e-103)))
		tmp = (((2.0 / ((k * tan(k)) * sin(k))) * l) / t) * (l / k);
	else
		tmp = 2.0 * (1.0 / ((k * (-k / l)) / (-1.0 / (t * (k / (l / k))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := If[Or[LessEqual[k, -5e-53], N[Not[LessEqual[k, 2.9e-103]], $MachinePrecision]], N[(N[(N[(N[(2.0 / N[(N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / t), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[(k * N[((-k) / l), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[(t * N[(k / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5e-53 or 2.8999999999999999e-103 < k

   1. Initial program 46.5

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Simplified 38.1

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 46.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 46.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 46.5	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 46.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 38.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 38.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 19.4

      \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}}} \]

   4. Simplified 19.4

      \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{\sin k \cdot t}}}} \]
      Proof

      [Start] 19.4	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.4	\[ \frac{2}{\tan k \cdot \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{\sin k \cdot t}}}} \]
      unpow2 [=>] 19.4	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{\sin k \cdot t}}} \]
      unpow2 [=>] 19.4	\[ \frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot t}}} \]

   5. Applied egg-rr 4.8

      \[\leadsto \frac{2}{\tan k \cdot \color{blue}{\left(\frac{k}{\frac{\ell}{\sin k}} \cdot \frac{k}{\frac{\ell}{t}}\right)}} \]

   6. Applied egg-rr 40.1

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)\right)} - 1}} \]

   7. Simplified 0.9

      \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}} \]
      Proof

      [Start] 40.1	\[ \frac{2}{e^{\mathsf{log1p}\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)\right)} - 1} \]
      expm1-def [=>] 29.6	\[ \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)\right)\right)}} \]
      expm1-log1p [=>] 4.5	\[ \frac{2}{\color{blue}{\left(k \cdot \frac{t}{\ell}\right) \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)}} \]
      associate-*r/ [=>] 4.8	\[ \frac{2}{\color{blue}{\frac{k \cdot t}{\ell}} \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)} \]
      *-commutative [=>] 4.8	\[ \frac{2}{\frac{\color{blue}{t \cdot k}}{\ell} \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)} \]
      associate-*r/ [<=] 0.9	\[ \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\ell}\right)} \cdot \left(\left(\tan k \cdot k\right) \cdot \frac{\sin k}{\ell}\right)} \]
      associate-*l* [=>] 0.9	\[ \frac{2}{\left(t \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\left(\tan k \cdot \left(k \cdot \frac{\sin k}{\ell}\right)\right)}} \]

   8. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\frac{2}{\left(k \cdot \tan k\right) \cdot \sin k} \cdot \ell}{t} \cdot \frac{\ell}{k}} \]

   if -5e-53 < k < 2.8999999999999999e-103

   1. Initial program 63.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   2. Simplified 54.7

      \[\leadsto \color{blue}{\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}} \cdot \frac{\ell}{{\left(\frac{k}{t}\right)}^{2}}} \]
      Proof

      [Start] 63.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 63.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 63.9	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 63.9	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 63.5	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 63.5	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 63.5	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 63.5	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 63.5	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 63.5	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 58.9

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 58.3

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]
      Proof

      [Start] 58.9	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 58.9	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 58.3	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 28.2

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot t}\right)} \]

   6. Applied egg-rr 19.6

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \left(\frac{k \cdot k}{\ell} \cdot t\right)}} \]

   7. Applied egg-rr 0.7

      \[\leadsto 2 \cdot \frac{1}{\color{blue}{\frac{\frac{k}{\ell} \cdot \left(-k\right)}{\frac{-1}{\frac{k}{\frac{\ell}{k}} \cdot t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-53} \lor \neg \left(k \leq 2.9 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{\frac{2}{\left(k \cdot \tan k\right) \cdot \sin k} \cdot \ell}{t} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{-k}{\ell}}{\frac{-1}{t \cdot \frac{k}{\frac{\ell}{k}}}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.8
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.12 \cdot 10^{-40} \lor \neg \left(k \leq 5 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k \cdot \frac{-k}{\ell}}{\frac{-1}{t \cdot \frac{k}{\frac{\ell}{k}}}}}\\ \end{array} \]

Alternative 2
Error	4.6
Cost	14025

\[\begin{array}{l} t_1 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+151} \lor \neg \left(t \leq 2.15 \cdot 10^{-98}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot \sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{\sin k \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]

Alternative 3
Error	4.3
Cost	13892

\[\begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{k \cdot \left(\tan k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)} \cdot \frac{\ell}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k \cdot \sin k}\right)\\ \end{array} \]

Alternative 4
Error	22.9
Cost	1280

\[2 \cdot \frac{1}{\frac{k \cdot \frac{-k}{\ell}}{\frac{-1}{t \cdot \frac{k}{\frac{\ell}{k}}}}} \]

Alternative 5
Error	25.3
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ t_2 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_2}\\ \mathbf{elif}\;t \leq 1.38 \cdot 10^{+249}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot t_1}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\ell}{t_2}\right)\\ \end{array} \]

Alternative 6
Error	22.9
Cost	1088

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ 2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Alternative 7
Error	25.9
Cost	960

\[2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \]

Alternative 8
Error	25.9
Cost	960

\[2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \]

Alternative 9
Error	26.5
Cost	960

\[2 \cdot \left(\ell \cdot \frac{\frac{\ell}{t \cdot \left(k \cdot k\right)}}{k \cdot k}\right) \]

Alternative 10
Error	25.6
Cost	960

\[2 \cdot \frac{\ell}{k \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \]

Alternative 11
Error	26.5
Cost	960

\[2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot k}}{k \cdot t} \]

Alternative 12
Error	24.2
Cost	960

\[2 \cdot \frac{\frac{\ell}{k}}{k \cdot \left(k \cdot \frac{k \cdot t}{\ell}\right)} \]

Alternative 13
Error	23.3
Cost	960

\[2 \cdot \frac{\frac{\ell}{k}}{k \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)} \]

Reproduce

herbie shell --seed 2023041 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))