Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 9.8

Time: 15.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := t \cdot \sqrt[3]{\sin k}\\ \mathbf{if}\;t \leq -4.306372628244695 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_3}{t_1} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{t_2}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.2695840797082967 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_3 \cdot \sqrt[3]{\tan k \cdot t_2}}{t_1}\right)}^{3}}\\ \end{array} \]

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 (cbrt l) 2.0))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (* t (cbrt (sin k)))))
   (if (<= t -4.306372628244695e-114)
     (/ 2.0 (pow (* (/ t_3 t_1) (* (cbrt (tan k)) (cbrt t_2))) 3.0))
     (if (<= t 2.2695840797082967e-123)
       (/
        2.0
        (*
         (/ (pow (sin k) 2.0) (* l l))
         (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))
       (/ 2.0 (pow (/ (* t_3 (cbrt (* (tan k) t_2))) t_1) 3.0))))))

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 = pow(cbrt(l), 2.0);
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = t * cbrt(sin(k));
	double tmp;
	if (t <= -4.306372628244695e-114) {
		tmp = 2.0 / pow(((t_3 / t_1) * (cbrt(tan(k)) * cbrt(t_2))), 3.0);
	} else if (t <= 2.2695840797082967e-123) {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	} else {
		tmp = 2.0 / pow(((t_3 * cbrt((tan(k) * t_2))) / t_1), 3.0);
	}
	return tmp;
}

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.cbrt(l), 2.0);
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = t * Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -4.306372628244695e-114) {
		tmp = 2.0 / Math.pow(((t_3 / t_1) * (Math.cbrt(Math.tan(k)) * Math.cbrt(t_2))), 3.0);
	} else if (t <= 2.2695840797082967e-123) {
		tmp = 2.0 / ((Math.pow(Math.sin(k), 2.0) / (l * l)) * (((t * (k * k)) / Math.cos(k)) + (2.0 * (Math.pow(t, 3.0) / Math.cos(k)))));
	} else {
		tmp = 2.0 / Math.pow(((t_3 * Math.cbrt((Math.tan(k) * t_2))) / t_1), 3.0);
	}
	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(l) ^ 2.0
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(t * cbrt(sin(k)))
	tmp = 0.0
	if (t <= -4.306372628244695e-114)
		tmp = Float64(2.0 / (Float64(Float64(t_3 / t_1) * Float64(cbrt(tan(k)) * cbrt(t_2))) ^ 3.0));
	elseif (t <= 2.2695840797082967e-123)
		tmp = Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / Float64(l * l)) * Float64(Float64(Float64(t * Float64(k * k)) / cos(k)) + Float64(2.0 * Float64((t ^ 3.0) / cos(k))))));
	else
		tmp = Float64(2.0 / (Float64(Float64(t_3 * cbrt(Float64(tan(k) * t_2))) / t_1) ^ 3.0));
	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[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.306372628244695e-114], N[(2.0 / N[Power[N[(N[(t$95$3 / t$95$1), $MachinePrecision] * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2695840797082967e-123], N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[t, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(N[(t$95$3 * N[Power[N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -4.306372628244695e-114

   1. Initial program 23.9

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

   2. Simplified 23.9

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

   3. Applied egg-rr 9.4

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

   4. Applied egg-rr 9.4

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

   5. Applied egg-rr 5.2

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

   6. Applied egg-rr 5.2

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

   if -4.306372628244695e-114 < t < 2.2695840797082967e-123

   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

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

   3. Taylor expanded in t around 0 41.4

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

   4. Simplified 26.3

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

   if 2.2695840797082967e-123 < t

   1. Initial program 25.5

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

   2. Simplified 25.5

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

   3. Applied egg-rr 9.7

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

   4. Applied egg-rr 9.7

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

   5. Applied egg-rr 5.6

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

   6. Applied egg-rr 5.6

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.306372628244695 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.2695840797082967 \cdot 10^{-123}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))))