Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 4.3

Time: 22.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \mathbf{if}\;t \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{\frac{{\sin k}^{2}}{\ell}} \cdot \frac{\frac{1}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (/
          (/ (* (* l 2.0) (/ (cos k) k)) k)
          (* (/ (sin k) l) (/ (sin k) (pow t -1.0))))))
   (if (<= t -8.4e+15)
     t_1
     (if (<= t 8.6e+49)
       (*
        (/ (/ (* l 2.0) (/ k (cos k))) (/ (pow (sin k) 2.0) l))
        (/ (/ 1.0 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 = (((l * 2.0) * (cos(k) / k)) / k) / ((sin(k) / l) * (sin(k) / pow(t, -1.0)));
	double tmp;
	if (t <= -8.4e+15) {
		tmp = t_1;
	} else if (t <= 8.6e+49) {
		tmp = (((l * 2.0) / (k / cos(k))) / (pow(sin(k), 2.0) / l)) * ((1.0 / 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 = (((l * 2.0d0) * (cos(k) / k)) / k) / ((sin(k) / l) * (sin(k) / (t ** (-1.0d0))))
    if (t <= (-8.4d+15)) then
        tmp = t_1
    else if (t <= 8.6d+49) then
        tmp = (((l * 2.0d0) / (k / cos(k))) / ((sin(k) ** 2.0d0) / l)) * ((1.0d0 / 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 = (((l * 2.0) * (Math.cos(k) / k)) / k) / ((Math.sin(k) / l) * (Math.sin(k) / Math.pow(t, -1.0)));
	double tmp;
	if (t <= -8.4e+15) {
		tmp = t_1;
	} else if (t <= 8.6e+49) {
		tmp = (((l * 2.0) / (k / Math.cos(k))) / (Math.pow(Math.sin(k), 2.0) / l)) * ((1.0 / 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 = (((l * 2.0) * (math.cos(k) / k)) / k) / ((math.sin(k) / l) * (math.sin(k) / math.pow(t, -1.0)))
	tmp = 0
	if t <= -8.4e+15:
		tmp = t_1
	elif t <= 8.6e+49:
		tmp = (((l * 2.0) / (k / math.cos(k))) / (math.pow(math.sin(k), 2.0) / l)) * ((1.0 / 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(Float64(Float64(Float64(l * 2.0) * Float64(cos(k) / k)) / k) / Float64(Float64(sin(k) / l) * Float64(sin(k) / (t ^ -1.0))))
	tmp = 0.0
	if (t <= -8.4e+15)
		tmp = t_1;
	elseif (t <= 8.6e+49)
		tmp = Float64(Float64(Float64(Float64(l * 2.0) / Float64(k / cos(k))) / Float64((sin(k) ^ 2.0) / l)) * Float64(Float64(1.0 / 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 = (((l * 2.0) * (cos(k) / k)) / k) / ((sin(k) / l) * (sin(k) / (t ^ -1.0)));
	tmp = 0.0;
	if (t <= -8.4e+15)
		tmp = t_1;
	elseif (t <= 8.6e+49)
		tmp = (((l * 2.0) / (k / cos(k))) / ((sin(k) ^ 2.0) / l)) * ((1.0 / 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[(N[(N[(N[(l * 2.0), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $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, -8.4e+15], t$95$1, If[LessEqual[t, 8.6e+49], N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.4e15 or 8.5999999999999998e49 < t

   1. Initial program 47.0

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

   2. Simplified 34.5

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

   3. Taylor expanded in t around 0 21.8

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

   4. Simplified 19.2

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

   5. Applied egg-rr 12.3

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

   6. Applied egg-rr 6.8

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

   7. Applied egg-rr 1.2

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

   if -8.4e15 < t < 8.5999999999999998e49

   1. Initial program 48.2

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

   2. Simplified 46.2

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

   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 25.7

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

   5. Applied egg-rr 16.9

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

   6. Applied egg-rr 13.5

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

   7. Applied egg-rr 7.5

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{\frac{k}{\cos k}}}{\frac{{\sin k}^{2}}{\ell}} \cdot \frac{\frac{1}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot 2\right) \cdot \frac{\cos k}{k}}{k}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{{t}^{-1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022181 
(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))))