Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 2.5

Time: 21.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-100}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (* 2.0 (/ (cos k) (/ (* (/ k l) (* (pow (sin k) 2.0) t)) (/ l k))))))
   (if (<= k -1e-150)
     t_1
     (if (<= k 1e-100)
       (* 2.0 (/ (cos k) (* (* (/ k l) (/ k l)) (* k (* k t)))))
       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 = 2.0 * (cos(k) / (((k / l) * (pow(sin(k), 2.0) * t)) / (l / k)));
	double tmp;
	if (k <= -1e-150) {
		tmp = t_1;
	} else if (k <= 1e-100) {
		tmp = 2.0 * (cos(k) / (((k / l) * (k / l)) * (k * (k * t))));
	} 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) :: tmp
    t_1 = 2.0d0 * (cos(k) / (((k / l) * ((sin(k) ** 2.0d0) * t)) / (l / k)))
    if (k <= (-1d-150)) then
        tmp = t_1
    else if (k <= 1d-100) then
        tmp = 2.0d0 * (cos(k) / (((k / l) * (k / l)) * (k * (k * t))))
    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 = 2.0 * (Math.cos(k) / (((k / l) * (Math.pow(Math.sin(k), 2.0) * t)) / (l / k)));
	double tmp;
	if (k <= -1e-150) {
		tmp = t_1;
	} else if (k <= 1e-100) {
		tmp = 2.0 * (Math.cos(k) / (((k / l) * (k / l)) * (k * (k * t))));
	} 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 = 2.0 * (math.cos(k) / (((k / l) * (math.pow(math.sin(k), 2.0) * t)) / (l / k)))
	tmp = 0
	if k <= -1e-150:
		tmp = t_1
	elif k <= 1e-100:
		tmp = 2.0 * (math.cos(k) / (((k / l) * (k / l)) * (k * (k * t))))
	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(2.0 * Float64(cos(k) / Float64(Float64(Float64(k / l) * Float64((sin(k) ^ 2.0) * t)) / Float64(l / k))))
	tmp = 0.0
	if (k <= -1e-150)
		tmp = t_1;
	elseif (k <= 1e-100)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(k * Float64(k * t)))));
	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 = 2.0 * (cos(k) / (((k / l) * ((sin(k) ^ 2.0) * t)) / (l / k)));
	tmp = 0.0;
	if (k <= -1e-150)
		tmp = t_1;
	elseif (k <= 1e-100)
		tmp = 2.0 * (cos(k) / (((k / l) * (k / l)) * (k * (k * t))));
	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[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(k / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1e-150], t$95$1, If[LessEqual[k, 1e-100], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if k < -1.00000000000000001e-150 or 1e-100 < k

   1. Initial program 46.6

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

   2. Simplified 38.4

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

   3. Taylor expanded in t around 0 20.2

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

   4. Simplified 19.8

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

   5. Applied egg-rr 7.5

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

   6. Applied egg-rr 1.6

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

   if -1.00000000000000001e-150 < k < 1e-100

   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

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

   3. Taylor expanded in t around 0 58.6

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

   4. Simplified 51.4

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

   5. Applied egg-rr 39.1

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

   6. Taylor expanded in k around 0 39.1

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

   7. Simplified 16.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{-150}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 10^{-100}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot t\right)}{\frac{\ell}{k}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022202 
(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))))