Toniolo and Linder, Equation (3a)

Average Error: 1.0 → 0.7

Time: 10.4s

Precision: binary64

Math

\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

↓

\[\begin{array}{l} t_0 := {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}\\ t_1 := {\sin kx}^{2}\\ t_2 := {\sin ky}^{2}\\ t_3 := t_1 + t_2\\ \mathbf{if}\;t_0 \cdot t_3 \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt[3]{{\left(\sqrt{\mathsf{fma}\left(t_0, t_3, 1\right)}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \frac{t_2}{Om \cdot Om} + \frac{t_1}{Om \cdot Om}\\ \sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot t_4}, 0.5 \cdot \frac{\sqrt{\frac{0.25}{t_4}}}{\ell}\right)}} \end{array}\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

↓

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (pow (/ (* 2.0 l) Om) 2.0))
        (t_1 (pow (sin kx) 2.0))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (+ t_1 t_2)))
   (if (<= (* t_0 t_3) INFINITY)
     (sqrt (+ 0.5 (/ 0.5 (cbrt (pow (sqrt (fma t_0 t_3 1.0)) 3.0)))))
     (let* ((t_4 (+ (/ t_2 (* Om Om)) (/ t_1 (* Om Om)))))
       (sqrt
        (+
         0.5
         (/
          0.5
          (fma l (sqrt (* 4.0 t_4)) (* 0.5 (/ (sqrt (/ 0.25 t_4)) l))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((1.0 / 2.0) * (1.0 + (1.0 / sqrt(1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))));
}

↓

double code(double l, double Om, double kx, double ky) {
	double t_0 = pow(((2.0 * l) / Om), 2.0);
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = t_1 + t_2;
	double tmp;
	if ((t_0 * t_3) <= ((double) INFINITY)) {
		tmp = sqrt(0.5 + (0.5 / cbrt(pow(sqrt(fma(t_0, t_3, 1.0)), 3.0))));
	} else {
		double t_4 = (t_2 / (Om * Om)) + (t_1 / (Om * Om));
		tmp = sqrt(0.5 + (0.5 / fma(l, sqrt(4.0 * t_4), (0.5 * (sqrt(0.25 / t_4) / l)))));
	}
	return tmp;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 2 l) Om) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))) < +inf.0

   1. Initial program 0.0

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Simplified 0.0

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

   3. Applied add-cbrt-cube_binary64 0.0

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)} \cdot \sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)}\right) \cdot \sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)}}}}} \]

   4. Simplified 0.0

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\sqrt[3]{\color{blue}{{\left(\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)}\right)}^{3}}}}} \]

   if +inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 2 l) Om) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))

   1. Initial program 64.0

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Simplified 64.0

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

   3. Taylor expanded in l around inf 40.7

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{0.5 \cdot \left(\frac{1}{\ell} \cdot \sqrt{\frac{1}{4 \cdot \frac{{\sin ky}^{2}}{{Om}^{2}} + 4 \cdot \frac{{\sin kx}^{2}}{{Om}^{2}}}}\right) + \ell \cdot \sqrt{4 \cdot \frac{{\sin ky}^{2}}{{Om}^{2}} + 4 \cdot \frac{{\sin kx}^{2}}{{Om}^{2}}}}}} \]

   4. Simplified 40.7

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot \left(\frac{{\sin ky}^{2}}{Om \cdot Om} + \frac{{\sin kx}^{2}}{Om \cdot Om}\right)}, 0.5 \cdot \frac{\sqrt{\frac{0.25}{\frac{{\sin ky}^{2}}{Om \cdot Om} + \frac{{\sin kx}^{2}}{Om \cdot Om}}}}{\ell}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt[3]{{\left(\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot \left(\frac{{\sin ky}^{2}}{Om \cdot Om} + \frac{{\sin kx}^{2}}{Om \cdot Om}\right)}, 0.5 \cdot \frac{\sqrt{\frac{0.25}{\frac{{\sin ky}^{2}}{Om \cdot Om} + \frac{{\sin kx}^{2}}{Om \cdot Om}}}}{\ell}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))