Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 5.2

Time: 23.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \cos k\right)}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \mathbf{if}\;t \leq -4886324.031650322:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-100}:\\ \;\;\;\;\frac{1}{\frac{{\sin k}^{2}}{\ell}} \cdot \frac{\cos k \cdot \frac{2}{k \cdot \frac{k}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* (/ (/ l k) k) (cos k)))
          (* (/ (sin k) l) (/ (sin k) (pow t -1.0))))))
   (if (<= t -4886324.031650322)
     t_1
     (if (<= t 1e-100)
       (*
        (/ 1.0 (/ (pow (sin k) 2.0) l))
        (/ (* (cos k) (/ 2.0 (* k (/ k l)))) 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 * (((l / k) / k) * cos(k))) / ((sin(k) / l) * (sin(k) / pow(t, -1.0)));
	double tmp;
	if (t <= -4886324.031650322) {
		tmp = t_1;
	} else if (t <= 1e-100) {
		tmp = (1.0 / (pow(sin(k), 2.0) / l)) * ((cos(k) * (2.0 / (k * (k / l)))) / 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 * (((l / k) / k) * cos(k))) / ((sin(k) / l) * (sin(k) / (t ** (-1.0d0))))
    if (t <= (-4886324.031650322d0)) then
        tmp = t_1
    else if (t <= 1d-100) then
        tmp = (1.0d0 / ((sin(k) ** 2.0d0) / l)) * ((cos(k) * (2.0d0 / (k * (k / l)))) / 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 * (((l / k) / k) * Math.cos(k))) / ((Math.sin(k) / l) * (Math.sin(k) / Math.pow(t, -1.0)));
	double tmp;
	if (t <= -4886324.031650322) {
		tmp = t_1;
	} else if (t <= 1e-100) {
		tmp = (1.0 / (Math.pow(Math.sin(k), 2.0) / l)) * ((Math.cos(k) * (2.0 / (k * (k / l)))) / 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 * (((l / k) / k) * math.cos(k))) / ((math.sin(k) / l) * (math.sin(k) / math.pow(t, -1.0)))
	tmp = 0
	if t <= -4886324.031650322:
		tmp = t_1
	elif t <= 1e-100:
		tmp = (1.0 / (math.pow(math.sin(k), 2.0) / l)) * ((math.cos(k) * (2.0 / (k * (k / l)))) / 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(Float64(2.0 * Float64(Float64(Float64(l / k) / k) * cos(k))) / Float64(Float64(sin(k) / l) * Float64(sin(k) / (t ^ -1.0))))
	tmp = 0.0
	if (t <= -4886324.031650322)
		tmp = t_1;
	elseif (t <= 1e-100)
		tmp = Float64(Float64(1.0 / Float64((sin(k) ^ 2.0) / l)) * Float64(Float64(cos(k) * Float64(2.0 / Float64(k * Float64(k / l)))) / 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 * (((l / k) / k) * cos(k))) / ((sin(k) / l) * (sin(k) / (t ^ -1.0)));
	tmp = 0.0;
	if (t <= -4886324.031650322)
		tmp = t_1;
	elseif (t <= 1e-100)
		tmp = (1.0 / ((sin(k) ^ 2.0) / l)) * ((cos(k) * (2.0 / (k * (k / l)))) / 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[(N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / N[Power[t, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4886324.031650322], t$95$1, If[LessEqual[t, 1e-100], N[(N[(1.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4886324.0316503216 or 1e-100 < t

   1. Initial program 43.2

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

   2. Simplified 31.0

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

   3. Taylor expanded in t around 0 20.8

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

   4. Simplified 18.9

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

   5. Applied egg-rr 11.6

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

   6. Taylor expanded in l around 0 11.5

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

   7. Simplified 6.4

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

   8. Applied egg-rr 1.9

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

   if -4886324.0316503216 < t < 1e-100

   1. Initial program 55.2

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

   2. Simplified 54.2

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

   3. Taylor expanded in t around 0 24.5

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

   4. Simplified 26.1

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

   5. Applied egg-rr 17.7

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

   6. Taylor expanded in l around 0 17.6

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

   7. Simplified 15.3

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

   8. Applied egg-rr 10.9

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4886324.031650322:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \cos k\right)}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \mathbf{elif}\;t \leq 10^{-100}:\\ \;\;\;\;\frac{1}{\frac{{\sin k}^{2}}{\ell}} \cdot \frac{\cos k \cdot \frac{2}{k \cdot \frac{k}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \cos k\right)}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022207 
(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))))