Toniolo and Linder, Equation (2)

Average Error: 9.8 → 1.0

Time: 14.9s

Precision: binary64

Math

\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ t_2 := {\left(\frac{Om}{Omc}\right)}^{2}\\ t_3 := \sqrt{1 - t_2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -9.45790398730271 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(t_3 \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.4570197606256265 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt{\frac{1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\\ \sin^{-1} \left(t_4 - 0.5 \cdot \left(t_2 \cdot t_4\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_3 \cdot t_1\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))

↓

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (* l (sqrt 0.5)) t))
        (t_2 (pow (/ Om Omc) 2.0))
        (t_3 (sqrt (- 1.0 t_2))))
   (if (<= (/ t l) -9.45790398730271e+156)
     (asin (* t_3 (- t_1)))
     (if (<= (/ t l) 2.4570197606256265e+121)
       (let* ((t_4 (sqrt (/ 1.0 (fma 2.0 (pow (/ t l) 2.0) 1.0)))))
         (asin (- t_4 (* 0.5 (* t_2 t_4)))))
       (asin (* t_3 t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(sqrt((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))));
}

↓

double code(double t, double l, double Om, double Omc) {
	double t_1 = (l * sqrt(0.5)) / t;
	double t_2 = pow((Om / Omc), 2.0);
	double t_3 = sqrt(1.0 - t_2);
	double tmp;
	if ((t / l) <= -9.45790398730271e+156) {
		tmp = asin(t_3 * -t_1);
	} else if ((t / l) <= 2.4570197606256265e+121) {
		double t_4 = sqrt(1.0 / fma(2.0, pow((t / l), 2.0), 1.0));
		tmp = asin(t_4 - (0.5 * (t_2 * t_4)));
	} else {
		tmp = asin(t_3 * t_1);
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -9.45790398730270958e156

   1. Initial program 34.0

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Simplified 34.0

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} \]

   3. Taylor expanded in t around -inf 8.2

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]

   4. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)} \]

   if -9.45790398730270958e156 < (/.f64 t l) < 2.45701976062562651e121

   1. Initial program 1.0

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Simplified 1.0

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} \]

   3. Taylor expanded in Om around 0 21.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1}} - 0.5 \cdot \left(\frac{{Om}^{2}}{{Omc}^{2}} \cdot \sqrt{\frac{1}{2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1}}\right)\right)} \]

   4. Simplified 1.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}} - 0.5 \cdot \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}} \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)} \]

   if 2.45701976062562651e121 < (/.f64 t l)

   1. Initial program 28.3

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Simplified 28.3

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} \]

   3. Taylor expanded in t around inf 7.4

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

   4. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -9.45790398730271 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.4570197606256265 \cdot 10^{+121}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}} - 0.5 \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021273 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))