Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 2.1

Time: 24.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;k \leq -2.808363570812293 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.0658959369000571 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot \left(\cos k \cdot \sqrt{{\sin k}^{-4}}\right)}{\frac{k}{\ell}}\\ \end{array} \]

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 (/ (/ l k) t)))
   (if (<= k -2.808363570812293e-144)
     (* 2.0 (/ (* t_1 (* (cos k) (pow (sin k) -2.0))) (/ k l)))
     (if (<= k 1.0658959369000571e-73)
       (* 2.0 (/ (/ (* (cos k) (/ (pow (/ l k) 2.0) t)) (sin k)) (sin k)))
       (* 2.0 (/ (* t_1 (* (cos k) (sqrt (pow (sin k) -4.0)))) (/ k 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 = (l / k) / t;
	double tmp;
	if (k <= -2.808363570812293e-144) {
		tmp = 2.0 * ((t_1 * (cos(k) * pow(sin(k), -2.0))) / (k / l));
	} else if (k <= 1.0658959369000571e-73) {
		tmp = 2.0 * (((cos(k) * (pow((l / k), 2.0) / t)) / sin(k)) / sin(k));
	} else {
		tmp = 2.0 * ((t_1 * (cos(k) * sqrt(pow(sin(k), -4.0)))) / (k / 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) :: tmp
    t_1 = (l / k) / t
    if (k <= (-2.808363570812293d-144)) then
        tmp = 2.0d0 * ((t_1 * (cos(k) * (sin(k) ** (-2.0d0)))) / (k / l))
    else if (k <= 1.0658959369000571d-73) then
        tmp = 2.0d0 * (((cos(k) * (((l / k) ** 2.0d0) / t)) / sin(k)) / sin(k))
    else
        tmp = 2.0d0 * ((t_1 * (cos(k) * sqrt((sin(k) ** (-4.0d0))))) / (k / 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 = (l / k) / t;
	double tmp;
	if (k <= -2.808363570812293e-144) {
		tmp = 2.0 * ((t_1 * (Math.cos(k) * Math.pow(Math.sin(k), -2.0))) / (k / l));
	} else if (k <= 1.0658959369000571e-73) {
		tmp = 2.0 * (((Math.cos(k) * (Math.pow((l / k), 2.0) / t)) / Math.sin(k)) / Math.sin(k));
	} else {
		tmp = 2.0 * ((t_1 * (Math.cos(k) * Math.sqrt(Math.pow(Math.sin(k), -4.0)))) / (k / 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 = (l / k) / t
	tmp = 0
	if k <= -2.808363570812293e-144:
		tmp = 2.0 * ((t_1 * (math.cos(k) * math.pow(math.sin(k), -2.0))) / (k / l))
	elif k <= 1.0658959369000571e-73:
		tmp = 2.0 * (((math.cos(k) * (math.pow((l / k), 2.0) / t)) / math.sin(k)) / math.sin(k))
	else:
		tmp = 2.0 * ((t_1 * (math.cos(k) * math.sqrt(math.pow(math.sin(k), -4.0)))) / (k / 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(l / k) / t)
	tmp = 0.0
	if (k <= -2.808363570812293e-144)
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(cos(k) * (sin(k) ^ -2.0))) / Float64(k / l)));
	elseif (k <= 1.0658959369000571e-73)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64((Float64(l / k) ^ 2.0) / t)) / sin(k)) / sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(t_1 * Float64(cos(k) * sqrt((sin(k) ^ -4.0)))) / Float64(k / 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 = (l / k) / t;
	tmp = 0.0;
	if (k <= -2.808363570812293e-144)
		tmp = 2.0 * ((t_1 * (cos(k) * (sin(k) ^ -2.0))) / (k / l));
	elseif (k <= 1.0658959369000571e-73)
		tmp = 2.0 * (((cos(k) * (((l / k) ^ 2.0) / t)) / sin(k)) / sin(k));
	else
		tmp = 2.0 * ((t_1 * (cos(k) * sqrt((sin(k) ^ -4.0)))) / (k / 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[(l / k), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[k, -2.808363570812293e-144], N[(2.0 * N[(N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0658959369000571e-73], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] * N[Sqrt[N[Power[N[Sin[k], $MachinePrecision], -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if k < -2.8083635708122931e-144

   1. Initial program 47.4

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

   2. Simplified 39.0

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

   3. Taylor expanded in t around 0 21.2

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

   4. Simplified 5.4

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

   5. Applied egg-rr 1.1

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

   if -2.8083635708122931e-144 < k < 1.0658959369000571e-73

   1. Initial program 63.8

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

   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 54.4

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

   4. Simplified 37.2

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

   5. Applied egg-rr 16.5

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

   if 1.0658959369000571e-73 < k

   1. Initial program 46.5

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

   2. Simplified 38.0

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

   3. Taylor expanded in t around 0 19.3

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

   4. Simplified 4.9

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

   5. Applied egg-rr 0.5

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

   6. Applied egg-rr 0.5

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.808363570812293 \cdot 10^{-144}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t} \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 1.0658959369000571 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}}{\sin k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{t} \cdot \left(\cos k \cdot \sqrt{{\sin k}^{-4}}\right)}{\frac{k}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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))))