Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 2.8

Time: 25.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{-2}\\ t_2 := t \cdot \frac{k}{\ell}\\ t_3 := \sqrt{\cos k \cdot \frac{\ell}{k \cdot t}} \cdot \frac{1}{\sin k}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot {\left(t_2 \cdot \frac{\frac{\frac{k}{\ell}}{t_1}}{\cos k}\right)}^{-1}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \left(t_3 \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{t_2}^{-1} \cdot \left(t_1 \cdot \cos k\right)}{\frac{k}{\ell}}\\ \end{array} \]

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 (pow (sin k) -2.0))
        (t_2 (* t (/ k l)))
        (t_3 (* (sqrt (* (cos k) (/ l (* k t)))) (/ 1.0 (sin k)))))
   (if (<= k -3e-124)
     (* 2.0 (pow (* t_2 (/ (/ (/ k l) t_1) (cos k))) -1.0))
     (if (<= k 7.5e-155)
       (* 2.0 (* t_3 (* t_3 (/ l k))))
       (* 2.0 (/ (* (pow t_2 -1.0) (* t_1 (cos k))) (/ 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 = pow(sin(k), -2.0);
	double t_2 = t * (k / l);
	double t_3 = sqrt((cos(k) * (l / (k * t)))) * (1.0 / sin(k));
	double tmp;
	if (k <= -3e-124) {
		tmp = 2.0 * pow((t_2 * (((k / l) / t_1) / cos(k))), -1.0);
	} else if (k <= 7.5e-155) {
		tmp = 2.0 * (t_3 * (t_3 * (l / k)));
	} else {
		tmp = 2.0 * ((pow(t_2, -1.0) * (t_1 * cos(k))) / (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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sin(k) ** (-2.0d0)
    t_2 = t * (k / l)
    t_3 = sqrt((cos(k) * (l / (k * t)))) * (1.0d0 / sin(k))
    if (k <= (-3d-124)) then
        tmp = 2.0d0 * ((t_2 * (((k / l) / t_1) / cos(k))) ** (-1.0d0))
    else if (k <= 7.5d-155) then
        tmp = 2.0d0 * (t_3 * (t_3 * (l / k)))
    else
        tmp = 2.0d0 * (((t_2 ** (-1.0d0)) * (t_1 * cos(k))) / (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 = Math.pow(Math.sin(k), -2.0);
	double t_2 = t * (k / l);
	double t_3 = Math.sqrt((Math.cos(k) * (l / (k * t)))) * (1.0 / Math.sin(k));
	double tmp;
	if (k <= -3e-124) {
		tmp = 2.0 * Math.pow((t_2 * (((k / l) / t_1) / Math.cos(k))), -1.0);
	} else if (k <= 7.5e-155) {
		tmp = 2.0 * (t_3 * (t_3 * (l / k)));
	} else {
		tmp = 2.0 * ((Math.pow(t_2, -1.0) * (t_1 * Math.cos(k))) / (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 = math.pow(math.sin(k), -2.0)
	t_2 = t * (k / l)
	t_3 = math.sqrt((math.cos(k) * (l / (k * t)))) * (1.0 / math.sin(k))
	tmp = 0
	if k <= -3e-124:
		tmp = 2.0 * math.pow((t_2 * (((k / l) / t_1) / math.cos(k))), -1.0)
	elif k <= 7.5e-155:
		tmp = 2.0 * (t_3 * (t_3 * (l / k)))
	else:
		tmp = 2.0 * ((math.pow(t_2, -1.0) * (t_1 * math.cos(k))) / (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 = sin(k) ^ -2.0
	t_2 = Float64(t * Float64(k / l))
	t_3 = Float64(sqrt(Float64(cos(k) * Float64(l / Float64(k * t)))) * Float64(1.0 / sin(k)))
	tmp = 0.0
	if (k <= -3e-124)
		tmp = Float64(2.0 * (Float64(t_2 * Float64(Float64(Float64(k / l) / t_1) / cos(k))) ^ -1.0));
	elseif (k <= 7.5e-155)
		tmp = Float64(2.0 * Float64(t_3 * Float64(t_3 * Float64(l / k))));
	else
		tmp = Float64(2.0 * Float64(Float64((t_2 ^ -1.0) * Float64(t_1 * cos(k))) / 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 = sin(k) ^ -2.0;
	t_2 = t * (k / l);
	t_3 = sqrt((cos(k) * (l / (k * t)))) * (1.0 / sin(k));
	tmp = 0.0;
	if (k <= -3e-124)
		tmp = 2.0 * ((t_2 * (((k / l) / t_1) / cos(k))) ^ -1.0);
	elseif (k <= 7.5e-155)
		tmp = 2.0 * (t_3 * (t_3 * (l / k)));
	else
		tmp = 2.0 * (((t_2 ^ -1.0) * (t_1 * cos(k))) / (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[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[Cos[k], $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -3e-124], N[(2.0 * N[Power[N[(t$95$2 * N[(N[(N[(k / l), $MachinePrecision] / t$95$1), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.5e-155], N[(2.0 * N[(t$95$3 * N[(t$95$3 * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[t$95$2, -1.0], $MachinePrecision] * N[(t$95$1 * N[Cos[k], $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 < -3e-124

   1. Initial program 46.4

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

   2. Simplified 38.3

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

   3. Taylor expanded in t around 0 20.0

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

   4. Simplified 5.2

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

   5. Applied egg-rr 0.8

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

   6. Applied egg-rr 1.2

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

   if -3e-124 < k < 7.5000000000000006e-155

   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

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

   3. Taylor expanded in t around 0 61.5

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

   4. Simplified 51.1

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

   5. Applied egg-rr 50.5

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

   6. Applied egg-rr 34.2

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

   if 7.5000000000000006e-155 < k

   1. Initial program 47.5

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

   2. Simplified 39.8

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

   3. Taylor expanded in t around 0 20.9

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

   4. Simplified 5.6

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

   5. Applied egg-rr 1.2

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

   6. Applied egg-rr 1.3

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot {\left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{\frac{\frac{k}{\ell}}{{\sin k}^{-2}}}{\cos k}\right)}^{-1}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;2 \cdot \left(\left(\sqrt{\cos k \cdot \frac{\ell}{k \cdot t}} \cdot \frac{1}{\sin k}\right) \cdot \left(\left(\sqrt{\cos k \cdot \frac{\ell}{k \cdot t}} \cdot \frac{1}{\sin k}\right) \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(t \cdot \frac{k}{\ell}\right)}^{-1} \cdot \left({\sin k}^{-2} \cdot \cos k\right)}{\frac{k}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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))))