Toniolo and Linder, Equation (10-)

Average Error: 47.3 → 12.9

Time: 22.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \cos k \cdot {k}^{-2}\\ t_3 := 2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot t_1}\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+152}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{\ell}{\sin k \cdot \sqrt{t}} \cdot \frac{\sqrt{2 \cdot \cos k}}{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-158}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 10^{-160}:\\ \;\;\;\;\left(2 \cdot t_2\right) \cdot \left(\ell \cdot \frac{\frac{\ell}{t}}{t_1}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(t_2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)}{\sqrt[3]{{\sin k}^{4}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\\ \end{array} \]

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 (* (cos k) (pow k -2.0)))
        (t_3 (* 2.0 (* (/ (cos k) k) (/ (* l l) (* (* k t) t_1))))))
   (if (<= l -1e+152)
     (pow
      (cbrt
       (pow (* (/ l (* (sin k) (sqrt t))) (/ (sqrt (* 2.0 (cos k))) k)) 2.0))
      3.0)
     (if (<= l -1e-158)
       t_3
       (if (<= l 1e-160)
         (* (* 2.0 t_2) (* l (/ (/ l t) t_1)))
         (if (<= l 1e+135)
           t_3
           (/
            (/ (* 2.0 (* t_2 (* l (/ l t)))) (cbrt (pow (sin k) 4.0)))
            (pow (cbrt (sin 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 = pow(sin(k), 2.0);
	double t_2 = cos(k) * pow(k, -2.0);
	double t_3 = 2.0 * ((cos(k) / k) * ((l * l) / ((k * t) * t_1)));
	double tmp;
	if (l <= -1e+152) {
		tmp = pow(cbrt(pow(((l / (sin(k) * sqrt(t))) * (sqrt((2.0 * cos(k))) / k)), 2.0)), 3.0);
	} else if (l <= -1e-158) {
		tmp = t_3;
	} else if (l <= 1e-160) {
		tmp = (2.0 * t_2) * (l * ((l / t) / t_1));
	} else if (l <= 1e+135) {
		tmp = t_3;
	} else {
		tmp = ((2.0 * (t_2 * (l * (l / t)))) / cbrt(pow(sin(k), 4.0))) / pow(cbrt(sin(k)), 2.0);
	}
	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.sin(k), 2.0);
	double t_2 = Math.cos(k) * Math.pow(k, -2.0);
	double t_3 = 2.0 * ((Math.cos(k) / k) * ((l * l) / ((k * t) * t_1)));
	double tmp;
	if (l <= -1e+152) {
		tmp = Math.pow(Math.cbrt(Math.pow(((l / (Math.sin(k) * Math.sqrt(t))) * (Math.sqrt((2.0 * Math.cos(k))) / k)), 2.0)), 3.0);
	} else if (l <= -1e-158) {
		tmp = t_3;
	} else if (l <= 1e-160) {
		tmp = (2.0 * t_2) * (l * ((l / t) / t_1));
	} else if (l <= 1e+135) {
		tmp = t_3;
	} else {
		tmp = ((2.0 * (t_2 * (l * (l / t)))) / Math.cbrt(Math.pow(Math.sin(k), 4.0))) / Math.pow(Math.cbrt(Math.sin(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 = sin(k) ^ 2.0
	t_2 = Float64(cos(k) * (k ^ -2.0))
	t_3 = Float64(2.0 * Float64(Float64(cos(k) / k) * Float64(Float64(l * l) / Float64(Float64(k * t) * t_1))))
	tmp = 0.0
	if (l <= -1e+152)
		tmp = cbrt((Float64(Float64(l / Float64(sin(k) * sqrt(t))) * Float64(sqrt(Float64(2.0 * cos(k))) / k)) ^ 2.0)) ^ 3.0;
	elseif (l <= -1e-158)
		tmp = t_3;
	elseif (l <= 1e-160)
		tmp = Float64(Float64(2.0 * t_2) * Float64(l * Float64(Float64(l / t) / t_1)));
	elseif (l <= 1e+135)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(t_2 * Float64(l * Float64(l / t)))) / cbrt((sin(k) ^ 4.0))) / (cbrt(sin(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], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[k], $MachinePrecision] * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+152], N[Power[N[Power[N[Power[N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[l, -1e-158], t$95$3, If[LessEqual[l, 1e-160], N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(l * N[(N[(l / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+135], t$95$3, N[(N[(N[(2.0 * N[(t$95$2 * N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[Sin[k], $MachinePrecision], 4.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -1e152

   1. Initial program 63.4

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

   2. Simplified 63.5

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

   3. Taylor expanded in t around 0 63.1

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

   4. Simplified 63.3

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

   5. Taylor expanded in l around 0 63.3

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

   6. Simplified 46.5

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

   7. Applied egg-rr 50.4

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

   if -1e152 < l < -1.00000000000000006e-158 or 9.9999999999999999e-161 < l < 9.99999999999999962e134

   1. Initial program 43.5

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

   2. Simplified 34.3

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

   3. Taylor expanded in t around 0 11.9

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

   4. Simplified 12.3

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

   5. Taylor expanded in l around 0 11.0

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

   6. Simplified 10.9

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

   7. Taylor expanded in l around 0 10.9

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

   8. Simplified 11.1

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

   9. Taylor expanded in k around inf 11.9

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

   10. Simplified 4.0

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

   if -1.00000000000000006e-158 < l < 9.9999999999999999e-161

   1. Initial program 45.0

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

   2. Simplified 36.2

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

   3. Taylor expanded in t around 0 18.2

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

   4. Simplified 17.9

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

   5. Taylor expanded in l around 0 18.0

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

   6. Simplified 12.2

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

   7. Taylor expanded in l around 0 12.2

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

   8. Simplified 8.7

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

   9. Applied egg-rr 8.7

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

   if 9.99999999999999962e134 < l

   1. Initial program 61.9

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

   2. Simplified 60.7

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

   3. Taylor expanded in t around 0 58.5

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

   4. Simplified 59.0

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

   5. Applied egg-rr 44.6

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+152}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\frac{\ell}{\sin k \cdot \sqrt{t}} \cdot \frac{\sqrt{2 \cdot \cos k}}{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-158}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 10^{-160}:\\ \;\;\;\;\left(2 \cdot \left(\cos k \cdot {k}^{-2}\right)\right) \cdot \left(\ell \cdot \frac{\frac{\ell}{t}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;\ell \leq 10^{+135}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{\left(k \cdot t\right) \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\left(\cos k \cdot {k}^{-2}\right) \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)}{\sqrt[3]{{\sin k}^{4}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(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))))