Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 13.2

Time: 11.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left(\frac{t}{\sqrt[3]{\cos k}} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot t_1\right)\right)}\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-104}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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
         (*
          l
          (*
           2.0
           (/
            l
            (fma
             2.0
             (pow (* (/ t (cbrt (cos k))) (pow (cbrt (sin k)) 2.0)) 3.0)
             (* (/ (* k k) (cos k)) (* t t_1))))))))
   (if (<= t -1.25e-52)
     t_2
     (if (<= t 7e-104)
       (* l (* (/ 2.0 (* k (* t k))) (/ (* l (cos k)) t_1)))
       t_2))))

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 = l * (2.0 * (l / fma(2.0, pow(((t / cbrt(cos(k))) * pow(cbrt(sin(k)), 2.0)), 3.0), (((k * k) / cos(k)) * (t * t_1)))));
	double tmp;
	if (t <= -1.25e-52) {
		tmp = t_2;
	} else if (t <= 7e-104) {
		tmp = l * ((2.0 / (k * (t * k))) * ((l * cos(k)) / t_1));
	} else {
		tmp = t_2;
	}
	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(l * Float64(2.0 * Float64(l / fma(2.0, (Float64(Float64(t / cbrt(cos(k))) * (cbrt(sin(k)) ^ 2.0)) ^ 3.0), Float64(Float64(Float64(k * k) / cos(k)) * Float64(t * t_1))))))
	tmp = 0.0
	if (t <= -1.25e-52)
		tmp = t_2;
	elseif (t <= 7e-104)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * cos(k)) / t_1)));
	else
		tmp = t_2;
	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[(l * N[(2.0 * N[(l / N[(2.0 * N[Power[N[(N[(t / N[Power[N[Cos[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] + N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.25e-52], t$95$2, If[LessEqual[t, 7e-104], N[(l * N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -1.25e-52 or 7.00000000000000057e-104 < t

   1. Initial program 22.9

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

   2. Simplified 18.7

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

   3. Taylor expanded in l around inf 23.1

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

   4. Simplified 23.1

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

   5. Applied egg-rr 13.7

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

   if -1.25e-52 < t < 7.00000000000000057e-104

   1. Initial program 59.0

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

   2. Simplified 59.1

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

   3. Taylor expanded in t around 0 23.3

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

   4. Simplified 11.8

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left(\frac{t}{\sqrt[3]{\cos k}} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-104}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\ell}{\mathsf{fma}\left(2, {\left(\frac{t}{\sqrt[3]{\cos k}} \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}\right)}^{3}, \frac{k \cdot k}{\cos k} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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))))