VandenBroeck and Keller, Equation (23)

Average Error: 13.8 → 0.2

Time: 12.4s

Precision: binary64

Math

\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{x}{F \cdot F} + \frac{1}{F \cdot F}\\ \mathbf{if}\;F \leq -94266079.00452884:\\ \;\;\;\;\frac{-1 + t_1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 50647081247.16809:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_1}{\sin B} - t_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

↓

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (+ (/ x (* F F)) (/ 1.0 (* F F)))))
   (if (<= F -94266079.00452884)
     (- (/ (+ -1.0 t_1) (sin B)) t_0)
     (if (<= F 50647081247.16809)
       (- (* F (/ 1.0 (/ (sin B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)))) t_0)
       (- (/ (- 1.0 t_1) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

↓

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (x / (F * F)) + (1.0 / (F * F));
	double tmp;
	if (F <= -94266079.00452884) {
		tmp = ((-1.0 + t_1) / sin(B)) - t_0;
	} else if (F <= 50647081247.16809) {
		tmp = (F * (1.0 / (sin(B) / pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)))) - t_0;
	} else {
		tmp = ((1.0 - t_1) / sin(B)) - t_0;
	}
	return tmp;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -94266079.0045288354

   1. Initial program 26.2

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Simplified 26.1

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 20.2

      \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   5. Simplified 20.2

      \[\leadsto \frac{\color{blue}{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded around -inf 0.2

      \[\leadsto \frac{\color{blue}{\left(\frac{x}{{F}^{2}} + \frac{1}{{F}^{2}}\right) - 1}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 0.2

      \[\leadsto \frac{\color{blue}{-1 + \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B} \]

   if -94266079.0045288354 < F < 50647081247.168091

   1. Initial program 0.4

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 0.3

      \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   5. Applied associate-/l*_binary64 0.3

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]

   6. Simplified 0.3

      \[\leadsto \frac{F}{\color{blue}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]
   7. Using strategy rm

   8. Applied div-inv_binary64 0.3

      \[\leadsto \color{blue}{F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]

   if 50647081247.168091 < F

   1. Initial program 25.5

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Simplified 25.5

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 19.4

      \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{\tan B} \]

   5. Simplified 19.4

      \[\leadsto \frac{\color{blue}{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded around inf 0.1

      \[\leadsto \frac{\color{blue}{1 - \left(\frac{x}{{F}^{2}} + \frac{1}{{F}^{2}}\right)}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 0.1

      \[\leadsto \frac{\color{blue}{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}}{\sin B} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -94266079.00452884:\\ \;\;\;\;\frac{-1 + \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50647081247.16809:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))