Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 12.6

Time: 13.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{\sqrt[3]{2}}{t \cdot t_1}\\ \mathbf{if}\;t \leq -1.1025396404998137 \cdot 10^{-29}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(\frac{{t_3}^{2}}{t_2} \cdot \frac{t_3}{\tan k}\right)\right)\\ \mathbf{elif}\;t \leq 1.3996591520002817 \cdot 10^{-64}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.627754681832356 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k}}{t_2 \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left({t_1}^{2} \cdot \frac{t}{\sqrt[3]{\cos k}}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {k}^{2}\right)\right)}\right)\\ \end{array} \]

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 (cbrt (sin k)))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (/ (cbrt 2.0) (* t t_1))))
   (if (<= t -1.1025396404998137e-29)
     (* l (* l (* (/ (pow t_3 2.0) t_2) (/ t_3 (tan k)))))
     (if (<= t 1.3996591520002817e-64)
       (* l (* (/ 2.0 (* k (* t k))) (/ (* l (cos k)) (pow (sin k) 2.0))))
       (if (<= t 5.627754681832356e+102)
         (* l (/ (* l (/ (/ 2.0 (pow t 3.0)) (sin k))) (* t_2 (tan k))))
         (*
          l
          (*
           2.0
           (/
            l
            (fma
             2.0
             (pow (* (pow t_1 2.0) (/ t (cbrt (cos k)))) 3.0)
             (* (/ (* k k) (cos k)) (* t (pow k 2.0))))))))))))

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 = cbrt(sin(k));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = cbrt(2.0) / (t * t_1);
	double tmp;
	if (t <= -1.1025396404998137e-29) {
		tmp = l * (l * ((pow(t_3, 2.0) / t_2) * (t_3 / tan(k))));
	} else if (t <= 1.3996591520002817e-64) {
		tmp = l * ((2.0 / (k * (t * k))) * ((l * cos(k)) / pow(sin(k), 2.0)));
	} else if (t <= 5.627754681832356e+102) {
		tmp = l * ((l * ((2.0 / pow(t, 3.0)) / sin(k))) / (t_2 * tan(k)));
	} else {
		tmp = l * (2.0 * (l / fma(2.0, pow((pow(t_1, 2.0) * (t / cbrt(cos(k)))), 3.0), (((k * k) / cos(k)) * (t * pow(k, 2.0))))));
	}
	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 = cbrt(sin(k))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(cbrt(2.0) / Float64(t * t_1))
	tmp = 0.0
	if (t <= -1.1025396404998137e-29)
		tmp = Float64(l * Float64(l * Float64(Float64((t_3 ^ 2.0) / t_2) * Float64(t_3 / tan(k)))));
	elseif (t <= 1.3996591520002817e-64)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * cos(k)) / (sin(k) ^ 2.0))));
	elseif (t <= 5.627754681832356e+102)
		tmp = Float64(l * Float64(Float64(l * Float64(Float64(2.0 / (t ^ 3.0)) / sin(k))) / Float64(t_2 * tan(k))));
	else
		tmp = Float64(l * Float64(2.0 * Float64(l / fma(2.0, (Float64((t_1 ^ 2.0) * Float64(t / cbrt(cos(k)))) ^ 3.0), Float64(Float64(Float64(k * k) / cos(k)) * Float64(t * (k ^ 2.0)))))));
	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[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[2.0, 1/3], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.1025396404998137e-29], N[(l * N[(l * N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(t$95$3 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3996591520002817e-64], N[(l * N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.627754681832356e+102], N[(l * N[(N[(l * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(2.0 * N[(l / N[(2.0 * N[Power[N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(t / N[Power[N[Cos[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if t < -1.102539640499814e-29

   1. Initial program 21.4

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

   2. Simplified 17.1

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

   3. Applied egg-rr 13.2

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

   if -1.102539640499814e-29 < t < 1.3996591520002817e-64

   1. Initial program 55.2

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

   2. Simplified 54.7

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

   3. Taylor expanded in t around 0 22.9

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

   4. Simplified 12.7

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

   if 1.3996591520002817e-64 < t < 5.6277546818323558e102

   1. Initial program 21.2

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

   2. Simplified 16.2

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

   3. Applied egg-rr 10.0

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

   if 5.6277546818323558e102 < t

   1. Initial program 24.2

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

   2. Simplified 21.0

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

   3. Taylor expanded in l around inf 26.9

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

   4. Simplified 26.9

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

   5. Applied egg-rr 13.2

      \[\leadsto \ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, \color{blue}{{\left({\left(\sqrt[3]{\sin k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\cos k}}\right)}^{3}}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right) \]

   6. Taylor expanded in k around 0 13.2

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1025396404998137 \cdot 10^{-29}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \left(\frac{{\left(\frac{\sqrt[3]{2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{2}}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{\sqrt[3]{2}}{t \cdot \sqrt[3]{\sin k}}}{\tan k}\right)\right)\\ \mathbf{elif}\;t \leq 1.3996591520002817 \cdot 10^{-64}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.627754681832356 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{\frac{2}{{t}^{3}}}{\sin k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left({\left(\sqrt[3]{\sin k}\right)}^{2} \cdot \frac{t}{\sqrt[3]{\cos k}}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {k}^{2}\right)\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))