Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 0.8

Time: 24.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\cos k \cdot \left(\ell \cdot 2\right)}{\frac{k}{{\sin k}^{-2}}}\\ \mathbf{if}\;k \leq -1.0153296323308793 \cdot 10^{-47}:\\ \;\;\;\;\left(\frac{1}{t} \cdot t_1\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;k \leq 3.306798165625435 \cdot 10^{-60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{k}, \frac{\ell}{k}, \ell \cdot -0.3333333333333333\right)}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \frac{k}{\ell}}\\ \end{array} \]

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 (/ (* (cos k) (* l 2.0)) (/ k (pow (sin k) -2.0)))))
   (if (<= k -1.0153296323308793e-47)
     (* (* (/ 1.0 t) t_1) (/ l k))
     (if (<= k 3.306798165625435e-60)
       (/
        (fma (/ 2.0 k) (/ l k) (* l -0.3333333333333333))
        (/ (* k t) (/ l k)))
       (/ t_1 (* t (/ 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 = (cos(k) * (l * 2.0)) / (k / pow(sin(k), -2.0));
	double tmp;
	if (k <= -1.0153296323308793e-47) {
		tmp = ((1.0 / t) * t_1) * (l / k);
	} else if (k <= 3.306798165625435e-60) {
		tmp = fma((2.0 / k), (l / k), (l * -0.3333333333333333)) / ((k * t) / (l / k));
	} else {
		tmp = t_1 / (t * (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(cos(k) * Float64(l * 2.0)) / Float64(k / (sin(k) ^ -2.0)))
	tmp = 0.0
	if (k <= -1.0153296323308793e-47)
		tmp = Float64(Float64(Float64(1.0 / t) * t_1) * Float64(l / k));
	elseif (k <= 3.306798165625435e-60)
		tmp = Float64(fma(Float64(2.0 / k), Float64(l / k), Float64(l * -0.3333333333333333)) / Float64(Float64(k * t) / Float64(l / k)));
	else
		tmp = Float64(t_1 / Float64(t * Float64(k / l)));
	end
	return 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[Cos[k], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] / N[(k / N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.0153296323308793e-47], N[(N[(N[(1.0 / t), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.306798165625435e-60], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / k), $MachinePrecision] + N[(l * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t * 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 < -1.0153296323308793e-47

   1. Initial program 45.2

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

   2. Simplified 36.6

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

   3. Taylor expanded in t around 0 18.6

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

   4. Simplified 10.7

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

   5. Applied egg-rr 5.1

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

   6. Applied egg-rr 3.9

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

   7. Applied egg-rr 0.4

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

   if -1.0153296323308793e-47 < k < 3.3067981656254352e-60

   1. Initial program 63.5

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

   2. Simplified 62.4

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

   3. Taylor expanded in t around 0 44.8

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

   4. Simplified 33.0

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

   5. Applied egg-rr 18.7

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

   6. Taylor expanded in k around 0 17.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{\ell}{{k}^{2}} - 0.3333333333333333 \cdot \ell}}{\frac{k \cdot t}{\frac{\ell}{k}}} \]

   7. Simplified 3.5

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{k}, \frac{\ell}{k}, \ell \cdot -0.3333333333333333\right)}}{\frac{k \cdot t}{\frac{\ell}{k}}} \]

   if 3.3067981656254352e-60 < 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 38.2

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

   3. Taylor expanded in t around 0 19.5

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

   4. Simplified 11.1

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

   5. Applied egg-rr 5.6

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

   6. Applied egg-rr 4.5

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

   7. Applied egg-rr 0.4

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.0153296323308793 \cdot 10^{-47}:\\ \;\;\;\;\left(\frac{1}{t} \cdot \frac{\cos k \cdot \left(\ell \cdot 2\right)}{\frac{k}{{\sin k}^{-2}}}\right) \cdot \frac{\ell}{k}\\ \mathbf{elif}\;k \leq 3.306798165625435 \cdot 10^{-60}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{k}, \frac{\ell}{k}, \ell \cdot -0.3333333333333333\right)}{\frac{k \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos k \cdot \left(\ell \cdot 2\right)}{\frac{k}{{\sin k}^{-2}}}}{t \cdot \frac{k}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(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))))