Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 14.9

Time: 33.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot t_1\\ t_3 := 0 \cdot t_2\\ t_4 := -t_3\\ t_5 := \mathsf{fma}\left(0, t_2, t_3\right)\\ t_6 := \frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)}\\ \mathbf{if}\;\ell \leq -3.0320863962966904 \cdot 10^{+184}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{1}{{\left(\frac{k}{\ell \cdot \sqrt{\cos k}}\right)}^{2}}}}\\ \mathbf{elif}\;\ell \leq -3.9559806813423186 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, t_6 \cdot \left(\mathsf{fma}\left(\sin k, \sin k \cdot t, -0 \cdot t\right) + \mathsf{fma}\left(0, t, 0 \cdot t\right)\right), t_4\right) + t_5}\\ \mathbf{elif}\;\ell \leq 2.4761954081248927 \cdot 10^{-234}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(k \cdot k\right) \cdot t_2}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 4.429858786323863 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(t_6 \cdot t_1\right) \cdot \left(k \cdot t\right) + t_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, {\left(\frac{\sqrt{\frac{k}{\cos k}}}{\ell}\right)}^{2} \cdot t_2, t_4\right) + t_5}\\ \end{array} \]

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 (* 0.0 t_2))
        (t_4 (- t_3))
        (t_5 (fma 0.0 t_2 t_3))
        (t_6 (/ k (* (cos k) (* l l)))))
   (if (<= l -3.0320863962966904e+184)
     (/ 2.0 (/ t_2 (/ 1.0 (pow (/ k (* l (sqrt (cos k)))) 2.0))))
     (if (<= l -3.9559806813423186e-156)
       (/
        2.0
        (+
         (fma
          k
          (*
           t_6
           (+ (fma (sin k) (* (sin k) t) (- (* 0.0 t))) (fma 0.0 t (* 0.0 t))))
          t_4)
         t_5))
       (if (<= l 2.4761954081248927e-234)
         (/ 2.0 (/ (/ (* (* k k) t_2) l) (* (cos k) l)))
         (if (<= l 4.429858786323863e+127)
           (/ 2.0 (+ (* (* t_6 t_1) (* k t)) t_5))
           (/
            2.0
            (+
             (fma k (* (pow (/ (sqrt (/ k (cos k))) l) 2.0) t_2) t_4)
             t_5))))))))

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 = 0.0 * t_2;
	double t_4 = -t_3;
	double t_5 = fma(0.0, t_2, t_3);
	double t_6 = k / (cos(k) * (l * l));
	double tmp;
	if (l <= -3.0320863962966904e+184) {
		tmp = 2.0 / (t_2 / (1.0 / pow((k / (l * sqrt(cos(k)))), 2.0)));
	} else if (l <= -3.9559806813423186e-156) {
		tmp = 2.0 / (fma(k, (t_6 * (fma(sin(k), (sin(k) * t), -(0.0 * t)) + fma(0.0, t, (0.0 * t)))), t_4) + t_5);
	} else if (l <= 2.4761954081248927e-234) {
		tmp = 2.0 / ((((k * k) * t_2) / l) / (cos(k) * l));
	} else if (l <= 4.429858786323863e+127) {
		tmp = 2.0 / (((t_6 * t_1) * (k * t)) + t_5);
	} else {
		tmp = 2.0 / (fma(k, (pow((sqrt((k / cos(k))) / l), 2.0) * t_2), t_4) + t_5);
	}
	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(0.0 * t_2)
	t_4 = Float64(-t_3)
	t_5 = fma(0.0, t_2, t_3)
	t_6 = Float64(k / Float64(cos(k) * Float64(l * l)))
	tmp = 0.0
	if (l <= -3.0320863962966904e+184)
		tmp = Float64(2.0 / Float64(t_2 / Float64(1.0 / (Float64(k / Float64(l * sqrt(cos(k)))) ^ 2.0))));
	elseif (l <= -3.9559806813423186e-156)
		tmp = Float64(2.0 / Float64(fma(k, Float64(t_6 * Float64(fma(sin(k), Float64(sin(k) * t), Float64(-Float64(0.0 * t))) + fma(0.0, t, Float64(0.0 * t)))), t_4) + t_5));
	elseif (l <= 2.4761954081248927e-234)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(k * k) * t_2) / l) / Float64(cos(k) * l)));
	elseif (l <= 4.429858786323863e+127)
		tmp = Float64(2.0 / Float64(Float64(Float64(t_6 * t_1) * Float64(k * t)) + t_5));
	else
		tmp = Float64(2.0 / Float64(fma(k, Float64((Float64(sqrt(Float64(k / cos(k))) / l) ^ 2.0) * t_2), t_4) + t_5));
	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[(0.0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(0.0 * t$95$2 + t$95$3), $MachinePrecision]}, Block[{t$95$6 = N[(k / N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.0320863962966904e+184], N[(2.0 / N[(t$95$2 / N[(1.0 / N[Power[N[(k / N[(l * N[Sqrt[N[Cos[k], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -3.9559806813423186e-156], N[(2.0 / N[(N[(k * N[(t$95$6 * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] + (-N[(0.0 * t), $MachinePrecision])), $MachinePrecision] + N[(0.0 * t + N[(0.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.4761954081248927e-234], N[(2.0 / N[(N[(N[(N[(k * k), $MachinePrecision] * t$95$2), $MachinePrecision] / l), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.429858786323863e+127], N[(2.0 / N[(N[(N[(t$95$6 * t$95$1), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(N[Power[N[(N[Sqrt[N[(k / N[Cos[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if l < -3.03208639629669038e184

   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

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

   3. Taylor expanded in t around 0 64.0

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

   4. Applied egg-rr 64.0

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

   5. Applied egg-rr 37.3

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {\sin k}^{2}}{\frac{1}{{\left(\frac{k}{\ell \cdot \sqrt{\cos k}}\right)}^{2}}}}} \]

   if -3.03208639629669038e184 < l < -3.95598068134231862e-156

   1. Initial program 46.1

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

   2. Simplified 37.9

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

   3. Taylor expanded in t around 0 16.4

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

   4. Applied egg-rr 14.8

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

   5. Applied egg-rr 8.0

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(k, \frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot \left(t \cdot {\sin k}^{2}\right), -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Applied egg-rr 7.6

      \[\leadsto \frac{2}{\mathsf{fma}\left(k, \frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\sin k, \sin k \cdot t, -0 \cdot t\right) + \mathsf{fma}\left(0, t, 0 \cdot t\right)\right)}, -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

   if -3.95598068134231862e-156 < l < 2.4761954081248927e-234

   1. Initial program 46.7

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

   2. Simplified 37.7

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

   3. Taylor expanded in t around 0 19.6

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

   4. Applied egg-rr 19.3

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

   5. Applied egg-rr 15.3

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

   if 2.4761954081248927e-234 < l < 4.42985878632386335e127

   1. Initial program 44.7

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

   2. Simplified 34.8

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

   3. Taylor expanded in t around 0 13.4

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

   4. Applied egg-rr 12.4

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

   5. Applied egg-rr 7.7

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(k, \frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot \left(t \cdot {\sin k}^{2}\right), -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Applied egg-rr 7.6

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot t\right)} + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]

   if 4.42985878632386335e127 < l

   1. Initial program 61.0

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

   2. Simplified 60.0

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

   3. Taylor expanded in t around 0 57.6

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

   4. Applied egg-rr 56.3

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

   5. Applied egg-rr 53.7

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(k, \frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot \left(t \cdot {\sin k}^{2}\right), -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]

   6. Applied egg-rr 46.0

      \[\leadsto \frac{2}{\mathsf{fma}\left(k, \color{blue}{{\left(\frac{\sqrt{\frac{k}{\cos k}}}{\ell}\right)}^{2}} \cdot \left(t \cdot {\sin k}^{2}\right), -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.0320863962966904 \cdot 10^{+184}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {\sin k}^{2}}{\frac{1}{{\left(\frac{k}{\ell \cdot \sqrt{\cos k}}\right)}^{2}}}}\\ \mathbf{elif}\;\ell \leq -3.9559806813423186 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, \frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot \left(\mathsf{fma}\left(\sin k, \sin k \cdot t, -0 \cdot t\right) + \mathsf{fma}\left(0, t, 0 \cdot t\right)\right), -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{elif}\;\ell \leq 2.4761954081248927 \cdot 10^{-234}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}}{\cos k \cdot \ell}}\\ \mathbf{elif}\;\ell \leq 4.429858786323863 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\cos k \cdot \left(\ell \cdot \ell\right)} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot t\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(k, {\left(\frac{\sqrt{\frac{k}{\cos k}}}{\ell}\right)}^{2} \cdot \left(t \cdot {\sin k}^{2}\right), -0 \cdot \left(t \cdot {\sin k}^{2}\right)\right) + \mathsf{fma}\left(0, t \cdot {\sin k}^{2}, 0 \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))