Toniolo and Linder, Equation (10+)

Average Error: 33.3 → 8.9

Time: 14.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot t_1\\ t_3 := 2 \cdot \left(\frac{\frac{\cos k}{t}}{t_1} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{+155}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{2 \cdot \frac{t_1 \cdot {t}^{3}}{\cos k} + \frac{{k}^{2} \cdot t_2}{\cos k}}\right)\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.62 \cdot 10^{+146}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, t_1 \cdot \frac{{t}^{3}}{\cos k}, t_2 \cdot \frac{k \cdot k}{\cos k}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (sin k) 2.0))
        (t_2 (* t t_1))
        (t_3 (* 2.0 (* (/ (/ (cos k) t) t_1) (* (/ l k) (/ l k))))))
   (if (<= k -4.5e+155)
     t_3
     (if (<= k -2.6e-58)
       (*
        l
        (*
         2.0
         (/
          l
          (+
           (* 2.0 (/ (* t_1 (pow t 3.0)) (cos k)))
           (/ (* (pow k 2.0) t_2) (cos k))))))
       (if (<= k 9.6e-70)
         (*
          l
          (pow
           (*
            (cbrt l)
            (/
             (/ (/ (cbrt 2.0) t) (cbrt (sin k)))
             (cbrt (* (+ 2.0 (pow (/ k t) 2.0)) (tan k)))))
           3.0))
         (if (<= k 1.62e+146)
           (*
            l
            (*
             2.0
             (/
              l
              (fma
               2.0
               (* t_1 (/ (pow t 3.0) (cos k)))
               (* t_2 (/ (* k k) (cos k)))))))
           t_3))))))

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 * t_1;
	double t_3 = 2.0 * (((cos(k) / t) / t_1) * ((l / k) * (l / k)));
	double tmp;
	if (k <= -4.5e+155) {
		tmp = t_3;
	} else if (k <= -2.6e-58) {
		tmp = l * (2.0 * (l / ((2.0 * ((t_1 * pow(t, 3.0)) / cos(k))) + ((pow(k, 2.0) * t_2) / cos(k)))));
	} else if (k <= 9.6e-70) {
		tmp = l * pow((cbrt(l) * (((cbrt(2.0) / t) / cbrt(sin(k))) / cbrt(((2.0 + pow((k / t), 2.0)) * tan(k))))), 3.0);
	} else if (k <= 1.62e+146) {
		tmp = l * (2.0 * (l / fma(2.0, (t_1 * (pow(t, 3.0) / cos(k))), (t_2 * ((k * k) / cos(k))))));
	} else {
		tmp = t_3;
	}
	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 * t_1)
	t_3 = Float64(2.0 * Float64(Float64(Float64(cos(k) / t) / t_1) * Float64(Float64(l / k) * Float64(l / k))))
	tmp = 0.0
	if (k <= -4.5e+155)
		tmp = t_3;
	elseif (k <= -2.6e-58)
		tmp = Float64(l * Float64(2.0 * Float64(l / Float64(Float64(2.0 * Float64(Float64(t_1 * (t ^ 3.0)) / cos(k))) + Float64(Float64((k ^ 2.0) * t_2) / cos(k))))));
	elseif (k <= 9.6e-70)
		tmp = Float64(l * (Float64(cbrt(l) * Float64(Float64(Float64(cbrt(2.0) / t) / cbrt(sin(k))) / cbrt(Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * tan(k))))) ^ 3.0));
	elseif (k <= 1.62e+146)
		tmp = Float64(l * Float64(2.0 * Float64(l / fma(2.0, Float64(t_1 * Float64((t ^ 3.0) / cos(k))), Float64(t_2 * Float64(Float64(k * k) / cos(k)))))));
	else
		tmp = t_3;
	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[(t * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.5e+155], t$95$3, If[LessEqual[k, -2.6e-58], N[(l * N[(2.0 * N[(l / N[(N[(2.0 * N[(N[(t$95$1 * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[k, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 9.6e-70], N[(l * N[Power[N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.62e+146], N[(l * N[(2.0 * N[(l / N[(2.0 * N[(t$95$1 * N[(N[Power[t, 3.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if k < -4.49999999999999973e155 or 1.62e146 < k

   1. Initial program 34.3

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

   2. Simplified 32.6

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

   3. Applied egg-rr 30.1

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

   4. Taylor expanded in t around 0 24.6

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

   5. Simplified 6.0

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

   if -4.49999999999999973e155 < k < -2.60000000000000007e-58

   1. Initial program 31.3

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

   2. Simplified 26.5

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

   3. Taylor expanded in l around inf 10.5

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

   if -2.60000000000000007e-58 < k < 9.6000000000000005e-70

   1. Initial program 35.2

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

   2. Simplified 33.0

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

   3. Applied egg-rr 22.0

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

   4. Applied egg-rr 10.3

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

   if 9.6000000000000005e-70 < k < 1.62e146

   1. Initial program 30.7

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

   2. Simplified 25.5

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

   3. Taylor expanded in l around inf 10.3

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

   4. Simplified 10.4

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.5 \cdot 10^{+155}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \mathbf{elif}\;k \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\cos k} + \frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}\right)\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{-70}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.62 \cdot 10^{+146}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\sin k}^{2} \cdot \frac{{t}^{3}}{\cos k}, \left(t \cdot {\sin k}^{2}\right) \cdot \frac{k \cdot k}{\cos k}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{t}}{{\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ \end{array} \]

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))))