Toniolo and Linder, Equation (2)

Average Error: 10.1 → 0.9

Time: 17.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4.240875889051068 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - t_1} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{4}\\ \mathbf{if}\;\frac{t}{\ell} \leq 1.3677435741571983 \cdot 10^{+22}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t_2}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right) \cdot \left(1 + t_1\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{t_2}}{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\\ \end{array} \]

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 (pow (/ Om Omc) 2.0)))
   (if (<= (/ t l) -4.240875889051068e+154)
     (asin (* (sqrt (- 1.0 t_1)) (- (/ (* l (sqrt 0.5)) t))))
     (let* ((t_2 (- 1.0 (pow (/ Om Omc) 4.0))))
       (if (<= (/ t l) 1.3677435741571983e+22)
         (asin
          (sqrt
           (log1p
            (expm1 (/ t_2 (* (fma 2.0 (pow (/ t l) 2.0) 1.0) (+ 1.0 t_1)))))))
         (asin
          (/
           (sqrt t_2)
           (* (hypot 1.0 (/ Om Omc)) (/ (* t (sqrt 2.0)) l)))))))))

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 = pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -4.240875889051068e+154) {
		tmp = asin(sqrt(1.0 - t_1) * -((l * sqrt(0.5)) / t));
	} else {
		double t_2 = 1.0 - pow((Om / Omc), 4.0);
		double tmp_1;
		if ((t / l) <= 1.3677435741571983e+22) {
			tmp_1 = asin(sqrt(log1p(expm1(t_2 / (fma(2.0, pow((t / l), 2.0), 1.0) * (1.0 + t_1))))));
		} else {
			tmp_1 = asin(sqrt(t_2) / (hypot(1.0, (Om / Omc)) * ((t * sqrt(2.0)) / l)));
		}
		tmp = tmp_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) < -4.240875889051068e154

   1. Initial program 34.2

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

   2. Simplified 34.2

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

   3. Taylor expanded in t around -inf 7.5

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

   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 -4.240875889051068e154 < (/.f64 t l) < 1.3677435741571983e22

   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. Applied flip--_binary64 1.0

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

   4. Applied associate-/l/_binary64 1.0

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

   5. Applied log1p-expm1-u_binary64 1.0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 \cdot 1 - {\left(\frac{Om}{Omc}\right)}^{2} \cdot {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right) \cdot \left(1 + {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)\right)}}\right) \]

   6. Simplified 1.0

      \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right) \cdot \left(1 + {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}\right)}\right) \]

   if 1.3677435741571983e22 < (/.f64 t l)

   1. Initial program 20.3

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

   2. Simplified 20.3

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

   3. Applied flip--_binary64 20.3

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

   4. Applied associate-/l/_binary64 20.3

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

   5. Applied sqrt-div_binary64 20.4

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

   6. Simplified 20.4

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

   7. Taylor expanded in t around inf 8.2

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

   8. Simplified 1.0

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\color{blue}{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{t \cdot \sqrt{2}}{\ell}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4.240875889051068 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.3677435741571983 \cdot 10^{+22}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right) \cdot \left(1 + {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022096 
(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)))))))