Toniolo and Linder, Equation (7)

Average Error: 43.0 → 10.9

Time: 17.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \sqrt{\mathsf{fma}\left(2, \frac{x}{x - 1}, \frac{2}{x - 1}\right)}\\ \mathbf{if}\;t \leq -1.0255551743284268 \cdot 10^{+42}:\\ \;\;\;\;\frac{t_1}{-t_2}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{t_1}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{if}\;t \leq -4.687014137306293 \cdot 10^{-169}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.9016161173507863 \cdot 10^{-280}:\\ \;\;\;\;\begin{array}{l} t_4 := 2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)\\ t_5 := \sqrt{\frac{1}{t_4}}\\ \frac{t_1}{-\mathsf{fma}\left(\frac{\ell \cdot \ell}{t \cdot \left(x \cdot x\right)}, t_5, \mathsf{fma}\left(t, \sqrt{t_4}, t_5 \cdot \left(\frac{\ell \cdot \ell}{t \cdot {x}^{3}} + \frac{\ell \cdot \ell}{t \cdot x}\right)\right)\right)} \end{array}\\ \mathbf{elif}\;t \leq 3.1941009868290845 \cdot 10^{+58}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_2}\\ \end{array}\\ \end{array} \]

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 (* t (sqrt 2.0)))
        (t_2 (* t (sqrt (fma 2.0 (/ x (- x 1.0)) (/ 2.0 (- x 1.0)))))))
   (if (<= t -1.0255551743284268e+42)
     (/ t_1 (- t_2))
     (let* ((t_3
             (/
              t_1
              (sqrt
               (fma 2.0 (+ (/ (* l l) x) (* t t)) (* 4.0 (/ (* t t) x)))))))
       (if (<= t -4.687014137306293e-169)
         t_3
         (if (<= t -2.9016161173507863e-280)
           (let* ((t_4
                   (+
                    2.0
                    (+ (/ 4.0 x) (+ (/ 4.0 (pow x 3.0)) (/ 4.0 (* x x))))))
                  (t_5 (sqrt (/ 1.0 t_4))))
             (/
              t_1
              (-
               (fma
                (/ (* l l) (* t (* x x)))
                t_5
                (fma
                 t
                 (sqrt t_4)
                 (*
                  t_5
                  (+ (/ (* l l) (* t (pow x 3.0))) (/ (* l l) (* t x)))))))))
           (if (<= t 3.1941009868290845e+58) t_3 (/ t_1 t_2))))))))

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 = t * sqrt(2.0);
	double t_2 = t * sqrt(fma(2.0, (x / (x - 1.0)), (2.0 / (x - 1.0))));
	double tmp;
	if (t <= -1.0255551743284268e+42) {
		tmp = t_1 / -t_2;
	} else {
		double t_3 = t_1 / sqrt(fma(2.0, (((l * l) / x) + (t * t)), (4.0 * ((t * t) / x))));
		double tmp_1;
		if (t <= -4.687014137306293e-169) {
			tmp_1 = t_3;
		} else if (t <= -2.9016161173507863e-280) {
			double t_4 = 2.0 + ((4.0 / x) + ((4.0 / pow(x, 3.0)) + (4.0 / (x * x))));
			double t_5 = sqrt(1.0 / t_4);
			tmp_1 = t_1 / -fma(((l * l) / (t * (x * x))), t_5, fma(t, sqrt(t_4), (t_5 * (((l * l) / (t * pow(x, 3.0))) + ((l * l) / (t * x))))));
		} else if (t <= 3.1941009868290845e+58) {
			tmp_1 = t_3;
		} else {
			tmp_1 = t_1 / t_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -1.02555517432842677e42

   1. Initial program 43.6

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

   2. Simplified 43.6

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

   3. Taylor expanded around -inf 4.1

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

   4. Simplified 4.1

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

   if -1.02555517432842677e42 < t < -4.6870141373062929e-169 or -2.90161611735078629e-280 < t < 3.19410098682908453e58

   1. Initial program 38.1

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

   2. Simplified 38.1

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

   3. Taylor expanded around inf 16.0

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

   4. Simplified 16.0

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

   if -4.6870141373062929e-169 < t < -2.90161611735078629e-280

   1. Initial program 63.0

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

   2. Simplified 63.0

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

   3. Taylor expanded around inf 41.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right)\right)}}} \]

   4. Simplified 41.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(4, \frac{t \cdot t}{{x}^{3}}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(\left(\frac{\ell \cdot \ell}{x} + t \cdot t\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right)\right)\right)\right)\right)}}} \]

   5. Taylor expanded around -inf 29.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\frac{{\ell}^{2}}{t \cdot {x}^{2}} \cdot \sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 4 \cdot \frac{1}{{x}^{2}}\right)\right)}} + \left(t \cdot \sqrt{2 + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 4 \cdot \frac{1}{{x}^{2}}\right)\right)} + \left(\frac{{\ell}^{2}}{t \cdot {x}^{3}} \cdot \sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 4 \cdot \frac{1}{{x}^{2}}\right)\right)}} + \frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 4 \cdot \frac{1}{{x}^{2}}\right)\right)}}\right)\right)\right)}} \]

   6. Simplified 29.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\mathsf{fma}\left(\frac{\ell \cdot \ell}{\left(x \cdot x\right) \cdot t}, \sqrt{\frac{1}{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}}, \mathsf{fma}\left(t, \sqrt{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}, \sqrt{\frac{1}{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}} \cdot \left(\frac{\ell \cdot \ell}{{x}^{3} \cdot t} + \frac{\ell \cdot \ell}{x \cdot t}\right)\right)\right)}} \]

   if 3.19410098682908453e58 < t

   1. Initial program 45.3

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

   2. Simplified 45.3

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

   3. Taylor expanded around inf 2.9

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

   4. Simplified 2.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\mathsf{fma}\left(2, \frac{x}{x - 1}, \frac{2}{x - 1}\right)}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.0255551743284268 \cdot 10^{+42}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\mathsf{fma}\left(2, \frac{x}{x - 1}, \frac{2}{x - 1}\right)}}\\ \mathbf{elif}\;t \leq -4.687014137306293 \cdot 10^{-169}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{elif}\;t \leq -2.9016161173507863 \cdot 10^{-280}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\mathsf{fma}\left(\frac{\ell \cdot \ell}{t \cdot \left(x \cdot x\right)}, \sqrt{\frac{1}{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}}, \mathsf{fma}\left(t, \sqrt{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}, \sqrt{\frac{1}{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}} \cdot \left(\frac{\ell \cdot \ell}{t \cdot {x}^{3}} + \frac{\ell \cdot \ell}{t \cdot x}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.1941009868290845 \cdot 10^{+58}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\mathsf{fma}\left(2, \frac{x}{x - 1}, \frac{2}{x - 1}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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)))))