VandenBroeck and Keller, Equation (23)

Average Error: 13.7 → 0.2

Time: 11.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -7.972631119662014 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 92747884.29196976:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \left(\frac{1}{F} - \left(\frac{x}{{F}^{3}} + \frac{1}{{F}^{3}}\right)\right)}{\sin B} - t_0\\ \end{array}\\ \end{array} \]

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
 (if (<= F -7.972631119662014e+64)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (let* ((t_0 (/ x (tan B))))
     (if (<= F 92747884.29196976)
       (- (* F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))) t_0)
       (-
        (/
         (* F (- (/ 1.0 F) (+ (/ x (pow F 3.0)) (/ 1.0 (pow F 3.0)))))
         (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 tmp;
	if (F <= -7.972631119662014e+64) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else {
		double t_0 = x / tan(B);
		double tmp_1;
		if (F <= 92747884.29196976) {
			tmp_1 = (F * (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
		} else {
			tmp_1 = ((F * ((1.0 / F) - ((x / pow(F, 3.0)) + (1.0 / pow(F, 3.0))))) / sin(B)) - t_0;
		}
		tmp = tmp_1;
	}
	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 < -7.97263111966201376e64

   1. Initial program 29.7

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

   2. Simplified 29.7

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

   3. Applied associate-*l/_binary64 22.9

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

   4. Taylor expanded in x around 0 23.0

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

   5. Taylor expanded in F around -inf 0.2

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x \cdot \cos B}{\sin B} \]

   if -7.97263111966201376e64 < F < 92747884.2919697613

   1. Initial program 0.6

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

   2. Simplified 0.5

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

   3. Applied div-inv_binary64 0.5

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

   4. Applied associate-*l*_binary64 0.3

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

   5. Simplified 0.3

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

   if 92747884.2919697613 < F

   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. Applied associate-*l/_binary64 20.4

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

   4. Taylor expanded in F around inf 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.972631119662014 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 92747884.29196976:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \left(\frac{1}{F} - \left(\frac{x}{{F}^{3}} + \frac{1}{{F}^{3}}\right)\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022004 
(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))))))