Toniolo and Linder, Equation (7)

Average Error: 43.3 → 9.3

Time: 12.2s

Precision: binary64

Math

\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

↓

\[\begin{array}{l} t_1 := {\left(\frac{\ell}{x}\right)}^{2}\\ t_2 := \frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)\\ t_3 := t \cdot \sqrt{2}\\ t_4 := \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}\\ t_5 := {\left(\frac{t}{x}\right)}^{2}\\ \mathbf{if}\;t \leq -0.008644579131846854:\\ \;\;\;\;\frac{t_3}{t_4 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -2.238472624987465 \cdot 10^{-206}:\\ \;\;\;\;\frac{t_3}{\sqrt{\mathsf{fma}\left(4, t_5, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, t_1, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -5.760693106007019 \cdot 10^{-274}:\\ \;\;\;\;\frac{t_3}{\left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right) \cdot \left(-\sqrt{\frac{1}{t_2}}\right) - t \cdot \sqrt{t_2}}\\ \mathbf{elif}\;t \leq 1.7217277470457817 \cdot 10^{+63}:\\ \;\;\;\;\frac{t_3}{{\left(\mathsf{fma}\left(4, t_5, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, t_1, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t \cdot t_4}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (pow (/ l x) 2.0))
        (t_2 (+ (/ 4.0 (* x x)) (+ 2.0 (/ 4.0 x))))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (sqrt (+ (/ 2.0 (+ x -1.0)) (* 2.0 (/ x (+ x -1.0))))))
        (t_5 (pow (/ t x) 2.0)))
   (if (<= t -0.008644579131846854)
     (/ t_3 (* t_4 (- t)))
     (if (<= t -2.238472624987465e-206)
       (/
        t_3
        (sqrt
         (fma
          4.0
          t_5
          (fma
           4.0
           (* t (/ t x))
           (fma 2.0 t_1 (* 2.0 (fma t t (* l (/ l x)))))))))
       (if (<= t -5.760693106007019e-274)
         (/
          t_3
          (-
           (*
            (+ (* (/ l x) (/ l t)) (/ (* (/ l x) (/ l x)) t))
            (- (sqrt (/ 1.0 t_2))))
           (* t (sqrt t_2))))
         (if (<= t 1.7217277470457817e+63)
           (/
            t_3
            (pow
             (fma
              4.0
              t_5
              (fma
               4.0
               (/ t (/ x t))
               (fma 2.0 t_1 (* 2.0 (fma t t (/ l (/ x l)))))))
             0.5))
           (/ t_3 (* t t_4))))))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

double code(double x, double l, double t) {
	double t_1 = pow((l / x), 2.0);
	double t_2 = (4.0 / (x * x)) + (2.0 + (4.0 / x));
	double t_3 = t * sqrt(2.0);
	double t_4 = sqrt(((2.0 / (x + -1.0)) + (2.0 * (x / (x + -1.0)))));
	double t_5 = pow((t / x), 2.0);
	double tmp;
	if (t <= -0.008644579131846854) {
		tmp = t_3 / (t_4 * -t);
	} else if (t <= -2.238472624987465e-206) {
		tmp = t_3 / sqrt(fma(4.0, t_5, fma(4.0, (t * (t / x)), fma(2.0, t_1, (2.0 * fma(t, t, (l * (l / x))))))));
	} else if (t <= -5.760693106007019e-274) {
		tmp = t_3 / (((((l / x) * (l / t)) + (((l / x) * (l / x)) / t)) * -sqrt((1.0 / t_2))) - (t * sqrt(t_2)));
	} else if (t <= 1.7217277470457817e+63) {
		tmp = t_3 / pow(fma(4.0, t_5, fma(4.0, (t / (x / t)), fma(2.0, t_1, (2.0 * fma(t, t, (l / (x / l))))))), 0.5);
	} else {
		tmp = t_3 / (t * t_4);
	}
	return tmp;
}

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(l / x) ^ 2.0
	t_2 = Float64(Float64(4.0 / Float64(x * x)) + Float64(2.0 + Float64(4.0 / x)))
	t_3 = Float64(t * sqrt(2.0))
	t_4 = sqrt(Float64(Float64(2.0 / Float64(x + -1.0)) + Float64(2.0 * Float64(x / Float64(x + -1.0)))))
	t_5 = Float64(t / x) ^ 2.0
	tmp = 0.0
	if (t <= -0.008644579131846854)
		tmp = Float64(t_3 / Float64(t_4 * Float64(-t)));
	elseif (t <= -2.238472624987465e-206)
		tmp = Float64(t_3 / sqrt(fma(4.0, t_5, fma(4.0, Float64(t * Float64(t / x)), fma(2.0, t_1, Float64(2.0 * fma(t, t, Float64(l * Float64(l / x)))))))));
	elseif (t <= -5.760693106007019e-274)
		tmp = Float64(t_3 / Float64(Float64(Float64(Float64(Float64(l / x) * Float64(l / t)) + Float64(Float64(Float64(l / x) * Float64(l / x)) / t)) * Float64(-sqrt(Float64(1.0 / t_2)))) - Float64(t * sqrt(t_2))));
	elseif (t <= 1.7217277470457817e+63)
		tmp = Float64(t_3 / (fma(4.0, t_5, fma(4.0, Float64(t / Float64(x / t)), fma(2.0, t_1, Float64(2.0 * fma(t, t, Float64(l / Float64(x / l))))))) ^ 0.5));
	else
		tmp = Float64(t_3 / Float64(t * t_4));
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[Power[N[(l / x), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(2.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(t / x), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -0.008644579131846854], N[(t$95$3 / N[(t$95$4 * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.238472624987465e-206], N[(t$95$3 / N[Sqrt[N[(4.0 * t$95$5 + N[(4.0 * N[(t * N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1 + N[(2.0 * N[(t * t + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.760693106007019e-274], N[(t$95$3 / N[(N[(N[(N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(l / x), $MachinePrecision] * N[(l / x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] - N[(t * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7217277470457817e+63], N[(t$95$3 / N[Power[N[(4.0 * t$95$5 + N[(4.0 * N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$1 + N[(2.0 * N[(t * t + N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(t$95$3 / N[(t * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -0.00864457913184685407

   1. Initial program 41.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 41.5

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   3. Taylor expanded in t around -inf 5.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-1 \cdot \left(\sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}} \cdot t\right)}} \]

   4. Simplified 5.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-t \cdot \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}}} \]

   if -0.00864457913184685407 < t < -2.2384726249874651e-206

   1. Initial program 37.4

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 37.4

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   3. Taylor expanded in x around inf 15.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 15.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Applied egg-rr 10.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left({\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)\right)}^{0.25}\right)}^{2}}} \]

   6. Applied egg-rr 10.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}}} \]

   if -2.2384726249874651e-206 < t < -5.7606931060070187e-274

   1. Initial program 62.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 62.2

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   3. Taylor expanded in x around inf 34.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 34.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Taylor expanded in t around -inf 28.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)}} + \left(\frac{{\ell}^{2}}{t \cdot {x}^{2}} \cdot \sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)}} + t \cdot \sqrt{2 + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)}\right)\right)}} \]

   6. Simplified 25.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{\frac{1}{\frac{4}{x \cdot x} + \left(\frac{4}{x} + 2\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right) + t \cdot \sqrt{\frac{4}{x \cdot x} + \left(\frac{4}{x} + 2\right)}\right)}} \]

   if -5.7606931060070187e-274 < t < 1.7217277470457817e63

   1. Initial program 42.8

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 42.8

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   3. Taylor expanded in x around inf 21.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)}}} \]

   4. Simplified 21.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)}}} \]

   5. Applied egg-rr 15.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)\right)}^{0.5}}} \]

   if 1.7217277470457817e63 < t

   1. Initial program 46.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   2. Simplified 46.1

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x + -1}, \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right), \ell \cdot \left(-\ell\right)\right)}}} \]

   3. Taylor expanded in t around inf 3.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}} \cdot t}} \]

   4. Simplified 3.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.008644579131846854:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -2.238472624987465 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, t \cdot \frac{t}{x}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \ell \cdot \frac{\ell}{x}\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -5.760693106007019 \cdot 10^{-274}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right) \cdot \left(-\sqrt{\frac{1}{\frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)}}\right) - t \cdot \sqrt{\frac{4}{x \cdot x} + \left(2 + \frac{4}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.7217277470457817 \cdot 10^{+63}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left(\mathsf{fma}\left(4, {\left(\frac{t}{x}\right)}^{2}, \mathsf{fma}\left(4, \frac{t}{\frac{x}{t}}, \mathsf{fma}\left(2, {\left(\frac{\ell}{x}\right)}^{2}, 2 \cdot \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right)\right)\right)\right)\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x + -1} + 2 \cdot \frac{x}{x + -1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022146 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))