Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 2.6

Time: 22.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot 2}{k} \cdot \left(\left({\sin k}^{-2} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\\ t_2 := \sqrt{\frac{\ell}{k}} \cdot \frac{1}{\sin k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-160}:\\ \;\;\;\;\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{\frac{1}{t_2} \cdot \frac{t}{t_2}}}{k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
 (let* ((t_1
         (* (/ (* l 2.0) k) (* (* (pow (sin k) -2.0) (/ l k)) (/ (cos k) t))))
        (t_2 (* (sqrt (/ l k)) (/ 1.0 (sin k)))))
   (if (<= k -1e-160)
     t_1
     (if (<= k 1e-160)
       (/ (* (* l 2.0) (/ (cos k) (* (/ 1.0 t_2) (/ t t_2)))) k)
       t_1))))

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 t_1 = ((l * 2.0) / k) * ((pow(sin(k), -2.0) * (l / k)) * (cos(k) / t));
	double t_2 = sqrt((l / k)) * (1.0 / sin(k));
	double tmp;
	if (k <= -1e-160) {
		tmp = t_1;
	} else if (k <= 1e-160) {
		tmp = ((l * 2.0) * (cos(k) / ((1.0 / t_2) * (t / t_2)))) / k;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((l * 2.0d0) / k) * (((sin(k) ** (-2.0d0)) * (l / k)) * (cos(k) / t))
    t_2 = sqrt((l / k)) * (1.0d0 / sin(k))
    if (k <= (-1d-160)) then
        tmp = t_1
    else if (k <= 1d-160) then
        tmp = ((l * 2.0d0) * (cos(k) / ((1.0d0 / t_2) * (t / t_2)))) / k
    else
        tmp = t_1
    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 t_1 = ((l * 2.0) / k) * ((Math.pow(Math.sin(k), -2.0) * (l / k)) * (Math.cos(k) / t));
	double t_2 = Math.sqrt((l / k)) * (1.0 / Math.sin(k));
	double tmp;
	if (k <= -1e-160) {
		tmp = t_1;
	} else if (k <= 1e-160) {
		tmp = ((l * 2.0) * (Math.cos(k) / ((1.0 / t_2) * (t / t_2)))) / k;
	} else {
		tmp = t_1;
	}
	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):
	t_1 = ((l * 2.0) / k) * ((math.pow(math.sin(k), -2.0) * (l / k)) * (math.cos(k) / t))
	t_2 = math.sqrt((l / k)) * (1.0 / math.sin(k))
	tmp = 0
	if k <= -1e-160:
		tmp = t_1
	elif k <= 1e-160:
		tmp = ((l * 2.0) * (math.cos(k) / ((1.0 / t_2) * (t / t_2)))) / k
	else:
		tmp = t_1
	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)
	t_1 = Float64(Float64(Float64(l * 2.0) / k) * Float64(Float64((sin(k) ^ -2.0) * Float64(l / k)) * Float64(cos(k) / t)))
	t_2 = Float64(sqrt(Float64(l / k)) * Float64(1.0 / sin(k)))
	tmp = 0.0
	if (k <= -1e-160)
		tmp = t_1;
	elseif (k <= 1e-160)
		tmp = Float64(Float64(Float64(l * 2.0) * Float64(cos(k) / Float64(Float64(1.0 / t_2) * Float64(t / t_2)))) / k);
	else
		tmp = t_1;
	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)
	t_1 = ((l * 2.0) / k) * (((sin(k) ^ -2.0) * (l / k)) * (cos(k) / t));
	t_2 = sqrt((l / k)) * (1.0 / sin(k));
	tmp = 0.0;
	if (k <= -1e-160)
		tmp = t_1;
	elseif (k <= 1e-160)
		tmp = ((l * 2.0) * (cos(k) / ((1.0 / t_2) * (t / t_2)))) / k;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(l / k), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1e-160], t$95$1, If[LessEqual[k, 1e-160], N[(N[(N[(l * 2.0), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(t / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if k < -9.9999999999999999e-161 or 9.9999999999999999e-161 < k

   1. Initial program 46.9

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

   2. Simplified 36.5

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

   3. Taylor expanded in k around inf 17.0

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

   4. Simplified 13.6

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

   5. Applied egg-rr 7.1

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

   6. Applied egg-rr 5.7

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

   7. Applied egg-rr 1.6

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

   if -9.9999999999999999e-161 < k < 9.9999999999999999e-161

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in k around inf 64.0

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

   4. Simplified 64.0

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

   5. Applied egg-rr 64.0

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

   6. Applied egg-rr 64.0

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

   7. Applied egg-rr 36.3

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

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \left(\left({\sin k}^{-2} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;k \leq 10^{-160}:\\ \;\;\;\;\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{\frac{1}{\sqrt{\frac{\ell}{k}} \cdot \frac{1}{\sin k}} \cdot \frac{t}{\sqrt{\frac{\ell}{k}} \cdot \frac{1}{\sin k}}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot 2}{k} \cdot \left(\left({\sin k}^{-2} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))))