Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 5.5

Time: 23.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot {\sin k}^{2}\\ t_2 := t \cdot t_1\\ \mathbf{if}\;t \leq -4.61396164152947 \cdot 10^{+63}:\\ \;\;\;\;\frac{\cos k \cdot \frac{2}{k}}{\frac{1}{\frac{\ell}{t_2}}}\\ \mathbf{elif}\;t \leq 10^{-90}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k \cdot 0.5}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t_1} \cdot \frac{\frac{2}{k}}{\frac{t}{\ell}}\\ \end{array} \]

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 (* (/ k l) (pow (sin k) 2.0))) (t_2 (* t t_1)))
   (if (<= t -4.61396164152947e+63)
     (/ (* (cos k) (/ 2.0 k)) (/ 1.0 (/ l t_2)))
     (if (<= t 1e-90)
       (* l (/ (/ (cos k) (* k 0.5)) t_2))
       (* (/ (cos k) t_1) (/ (/ 2.0 k) (/ t l)))))))

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 = (k / l) * pow(sin(k), 2.0);
	double t_2 = t * t_1;
	double tmp;
	if (t <= -4.61396164152947e+63) {
		tmp = (cos(k) * (2.0 / k)) / (1.0 / (l / t_2));
	} else if (t <= 1e-90) {
		tmp = l * ((cos(k) / (k * 0.5)) / t_2);
	} else {
		tmp = (cos(k) / t_1) * ((2.0 / k) / (t / l));
	}
	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 = (k / l) * (sin(k) ** 2.0d0)
    t_2 = t * t_1
    if (t <= (-4.61396164152947d+63)) then
        tmp = (cos(k) * (2.0d0 / k)) / (1.0d0 / (l / t_2))
    else if (t <= 1d-90) then
        tmp = l * ((cos(k) / (k * 0.5d0)) / t_2)
    else
        tmp = (cos(k) / t_1) * ((2.0d0 / k) / (t / l))
    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 = (k / l) * Math.pow(Math.sin(k), 2.0);
	double t_2 = t * t_1;
	double tmp;
	if (t <= -4.61396164152947e+63) {
		tmp = (Math.cos(k) * (2.0 / k)) / (1.0 / (l / t_2));
	} else if (t <= 1e-90) {
		tmp = l * ((Math.cos(k) / (k * 0.5)) / t_2);
	} else {
		tmp = (Math.cos(k) / t_1) * ((2.0 / k) / (t / l));
	}
	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 = (k / l) * math.pow(math.sin(k), 2.0)
	t_2 = t * t_1
	tmp = 0
	if t <= -4.61396164152947e+63:
		tmp = (math.cos(k) * (2.0 / k)) / (1.0 / (l / t_2))
	elif t <= 1e-90:
		tmp = l * ((math.cos(k) / (k * 0.5)) / t_2)
	else:
		tmp = (math.cos(k) / t_1) * ((2.0 / k) / (t / l))
	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(k / l) * (sin(k) ^ 2.0))
	t_2 = Float64(t * t_1)
	tmp = 0.0
	if (t <= -4.61396164152947e+63)
		tmp = Float64(Float64(cos(k) * Float64(2.0 / k)) / Float64(1.0 / Float64(l / t_2)));
	elseif (t <= 1e-90)
		tmp = Float64(l * Float64(Float64(cos(k) / Float64(k * 0.5)) / t_2));
	else
		tmp = Float64(Float64(cos(k) / t_1) * Float64(Float64(2.0 / k) / Float64(t / l)));
	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 = (k / l) * (sin(k) ^ 2.0);
	t_2 = t * t_1;
	tmp = 0.0;
	if (t <= -4.61396164152947e+63)
		tmp = (cos(k) * (2.0 / k)) / (1.0 / (l / t_2));
	elseif (t <= 1e-90)
		tmp = l * ((cos(k) / (k * 0.5)) / t_2);
	else
		tmp = (cos(k) / t_1) * ((2.0 / k) / (t / l));
	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[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * t$95$1), $MachinePrecision]}, If[LessEqual[t, -4.61396164152947e+63], N[(N[(N[Cos[k], $MachinePrecision] * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-90], N[(l * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * 0.5), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.61396164152947014e63

   1. Initial program 49.6

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

   2. Simplified 36.7

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

   3. Taylor expanded in t around 0 22.3

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

   4. Simplified 19.5

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

   5. Applied egg-rr 12.6

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

   6. Taylor expanded in k around inf 19.5

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

   7. Simplified 9.0

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

   8. Applied egg-rr 7.4

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

   if -4.61396164152947014e63 < t < 9.99999999999999995e-91

   1. Initial program 51.6

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

   2. Simplified 49.5

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

   3. Taylor expanded in t around 0 23.7

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

   4. Simplified 25.1

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

   5. Applied egg-rr 17.2

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

   6. Taylor expanded in k around inf 17.2

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

   7. Simplified 9.3

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

   8. Applied egg-rr 5.1

      \[\leadsto \color{blue}{\frac{\frac{\cos k}{k \cdot 0.5}}{t \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)} \cdot \ell} \]

   if 9.99999999999999995e-91 < t

   1. Initial program 40.5

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

   2. Simplified 28.9

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

   3. Taylor expanded in t around 0 20.6

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

   4. Simplified 18.6

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

   5. Applied egg-rr 12.3

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

   6. Taylor expanded in k around inf 17.3

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

   7. Simplified 5.8

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

   8. Applied egg-rr 4.5

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.61396164152947 \cdot 10^{+63}:\\ \;\;\;\;\frac{\cos k \cdot \frac{2}{k}}{\frac{1}{\frac{\ell}{t \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)}}}\\ \mathbf{elif}\;t \leq 10^{-90}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k \cdot 0.5}}{t \cdot \left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{\frac{k}{\ell} \cdot {\sin k}^{2}} \cdot \frac{\frac{2}{k}}{\frac{t}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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))))