Toniolo and Linder, Equation (10-)

Average Error: 47.7 → 9.7

Time: 22.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \cos k \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{1}{t_2 \cdot t}\\ t_4 := 2 \cdot \left(\left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\right) \cdot t_3\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+199}:\\ \;\;\;\;2 \cdot \frac{t_3}{\frac{1}{t_1}}\\ \mathbf{elif}\;\ell \leq -6.728547701870032 \cdot 10^{-158}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 1.2884415081700598 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{\frac{t_1}{t_2}}{t}\\ \mathbf{elif}\;\ell \leq 10^{-22}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \sin k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (* (cos k) (/ l (* k (/ k l)))))
        (t_2 (pow (sin k) 2.0))
        (t_3 (/ 1.0 (* t_2 t)))
        (t_4 (* 2.0 (* (* (cos k) (/ (* (/ l k) l) k)) t_3))))
   (if (<= l -1e+199)
     (* 2.0 (/ t_3 (/ 1.0 t_1)))
     (if (<= l -6.728547701870032e-158)
       t_4
       (if (<= l 1.2884415081700598e-132)
         (* 2.0 (/ (/ t_1 t_2) t))
         (if (<= l 1e-22)
           (/ (/ 2.0 (tan k)) (/ (* t (* (* k k) (sin k))) (* l l)))
           t_4))))))

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 = cos(k) * (l / (k * (k / l)));
	double t_2 = pow(sin(k), 2.0);
	double t_3 = 1.0 / (t_2 * t);
	double t_4 = 2.0 * ((cos(k) * (((l / k) * l) / k)) * t_3);
	double tmp;
	if (l <= -1e+199) {
		tmp = 2.0 * (t_3 / (1.0 / t_1));
	} else if (l <= -6.728547701870032e-158) {
		tmp = t_4;
	} else if (l <= 1.2884415081700598e-132) {
		tmp = 2.0 * ((t_1 / t_2) / t);
	} else if (l <= 1e-22) {
		tmp = (2.0 / tan(k)) / ((t * ((k * k) * sin(k))) / (l * l));
	} else {
		tmp = t_4;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = cos(k) * (l / (k * (k / l)))
    t_2 = sin(k) ** 2.0d0
    t_3 = 1.0d0 / (t_2 * t)
    t_4 = 2.0d0 * ((cos(k) * (((l / k) * l) / k)) * t_3)
    if (l <= (-1d+199)) then
        tmp = 2.0d0 * (t_3 / (1.0d0 / t_1))
    else if (l <= (-6.728547701870032d-158)) then
        tmp = t_4
    else if (l <= 1.2884415081700598d-132) then
        tmp = 2.0d0 * ((t_1 / t_2) / t)
    else if (l <= 1d-22) then
        tmp = (2.0d0 / tan(k)) / ((t * ((k * k) * sin(k))) / (l * l))
    else
        tmp = t_4
    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 = Math.cos(k) * (l / (k * (k / l)));
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = 1.0 / (t_2 * t);
	double t_4 = 2.0 * ((Math.cos(k) * (((l / k) * l) / k)) * t_3);
	double tmp;
	if (l <= -1e+199) {
		tmp = 2.0 * (t_3 / (1.0 / t_1));
	} else if (l <= -6.728547701870032e-158) {
		tmp = t_4;
	} else if (l <= 1.2884415081700598e-132) {
		tmp = 2.0 * ((t_1 / t_2) / t);
	} else if (l <= 1e-22) {
		tmp = (2.0 / Math.tan(k)) / ((t * ((k * k) * Math.sin(k))) / (l * l));
	} else {
		tmp = t_4;
	}
	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 = math.cos(k) * (l / (k * (k / l)))
	t_2 = math.pow(math.sin(k), 2.0)
	t_3 = 1.0 / (t_2 * t)
	t_4 = 2.0 * ((math.cos(k) * (((l / k) * l) / k)) * t_3)
	tmp = 0
	if l <= -1e+199:
		tmp = 2.0 * (t_3 / (1.0 / t_1))
	elif l <= -6.728547701870032e-158:
		tmp = t_4
	elif l <= 1.2884415081700598e-132:
		tmp = 2.0 * ((t_1 / t_2) / t)
	elif l <= 1e-22:
		tmp = (2.0 / math.tan(k)) / ((t * ((k * k) * math.sin(k))) / (l * l))
	else:
		tmp = t_4
	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(cos(k) * Float64(l / Float64(k * Float64(k / l))))
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(1.0 / Float64(t_2 * t))
	t_4 = Float64(2.0 * Float64(Float64(cos(k) * Float64(Float64(Float64(l / k) * l) / k)) * t_3))
	tmp = 0.0
	if (l <= -1e+199)
		tmp = Float64(2.0 * Float64(t_3 / Float64(1.0 / t_1)));
	elseif (l <= -6.728547701870032e-158)
		tmp = t_4;
	elseif (l <= 1.2884415081700598e-132)
		tmp = Float64(2.0 * Float64(Float64(t_1 / t_2) / t));
	elseif (l <= 1e-22)
		tmp = Float64(Float64(2.0 / tan(k)) / Float64(Float64(t * Float64(Float64(k * k) * sin(k))) / Float64(l * l)));
	else
		tmp = t_4;
	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 = cos(k) * (l / (k * (k / l)));
	t_2 = sin(k) ^ 2.0;
	t_3 = 1.0 / (t_2 * t);
	t_4 = 2.0 * ((cos(k) * (((l / k) * l) / k)) * t_3);
	tmp = 0.0;
	if (l <= -1e+199)
		tmp = 2.0 * (t_3 / (1.0 / t_1));
	elseif (l <= -6.728547701870032e-158)
		tmp = t_4;
	elseif (l <= 1.2884415081700598e-132)
		tmp = 2.0 * ((t_1 / t_2) / t);
	elseif (l <= 1e-22)
		tmp = (2.0 / tan(k)) / ((t * ((k * k) * sin(k))) / (l * l));
	else
		tmp = t_4;
	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[Cos[k], $MachinePrecision] * N[(l / N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(t$95$2 * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+199], N[(2.0 * N[(t$95$3 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -6.728547701870032e-158], t$95$4, If[LessEqual[l, 1.2884415081700598e-132], N[(2.0 * N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e-22], N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[(t * N[(N[(k * k), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.0000000000000001e199

   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

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

   3. Taylor expanded in k around inf 64.0

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

   4. Applied egg-rr 64.0

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

   5. Taylor expanded in l around 0 64.0

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

   6. Simplified 50.9

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

   7. Applied egg-rr 15.8

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{1}{{\sin k}^{2} \cdot t}}{\frac{1}{\cos k \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}}}} \]

   if -1.0000000000000001e199 < l < -6.72854770187003164e-158 or 1e-22 < l

   1. Initial program 48.3

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

   2. Simplified 41.6

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

   3. Taylor expanded in k around inf 24.4

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

   4. Applied egg-rr 23.0

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

   5. Taylor expanded in l around 0 22.9

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

   6. Simplified 18.4

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

   7. Applied egg-rr 9.0

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

   if -6.72854770187003164e-158 < l < 1.2884415081700598e-132

   1. Initial program 45.9

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

   2. Simplified 37.0

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

   3. Taylor expanded in k around inf 18.8

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

   4. Applied egg-rr 18.8

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

   5. Taylor expanded in l around 0 18.8

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

   6. Simplified 13.3

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

   7. Applied egg-rr 10.7

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

   if 1.2884415081700598e-132 < l < 1e-22

   1. Initial program 42.5

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

   2. Simplified 31.3

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

   3. Taylor expanded in k around inf 4.3

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

   4. Simplified 6.9

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \sin k\right)}{\ell \cdot \ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+199}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{{\sin k}^{2} \cdot t}}{\frac{1}{\cos k \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}}}\\ \mathbf{elif}\;\ell \leq -6.728547701870032 \cdot 10^{-158}:\\ \;\;\;\;2 \cdot \left(\left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\right) \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)\\ \mathbf{elif}\;\ell \leq 1.2884415081700598 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}}{{\sin k}^{2}}}{t}\\ \mathbf{elif}\;\ell \leq 10^{-22}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{t \cdot \left(\left(k \cdot k\right) \cdot \sin k\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \ell}{k}\right) \cdot \frac{1}{{\sin k}^{2} \cdot t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(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))))