VandenBroeck and Keller, Equation (23)

Average Error: 14.0 → 0.3

Time: 10.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -7.39940744743415903 \cdot 10^{52}:\\ \;\;\;\;\frac{\frac{\frac{1}{F}}{F} - 1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 244697.137374221085:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{1}{\frac{\tan B}{x \cdot 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

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 VAR;
	if ((F <= -7.399407447434159e+52)) {
		VAR = (((((1.0 / F) / F) - 1.0) / sin(B)) - (x * (1.0 / tan(B))));
	} else {
		double VAR_1;
		if ((F <= 244697.13737422109)) {
			VAR_1 = (((F * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0))) / sin(B)) - (1.0 / (tan(B) / (x * 1.0))));
		} else {
			VAR_1 = (((1.0 - ((1.0 / F) / F)) / sin(B)) - ((x * 1.0) / tan(B)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -7.399407447434159e+52

   1. Initial program 29.2

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

   2. Simplified 29.2

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}}\]
   3. Using strategy rm

   4. Applied associate-*l/ 23.0

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

   5. Taylor expanded around -inf 0.2

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{{F}^{2}} - 1}}{\sin B} - x \cdot \frac{1}{\tan B}\]

   6. Simplified 0.2

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

   if -7.399407447434159e+52 < F < 244697.13737422109

   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 0.6

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}}\]
   3. Using strategy rm

   4. Applied associate-*l/ 0.4

      \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B}\]
   5. Using strategy rm

   6. Applied associate-*r/ 0.3

      \[\leadsto \frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \color{blue}{\frac{x \cdot 1}{\tan B}}\]
   7. Using strategy rm

   8. Applied clear-num 0.4

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

   if 244697.13737422109 < F

   1. Initial program 25.7

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

   2. Simplified 25.7

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}}\]
   3. Using strategy rm

   4. Applied associate-*l/ 19.4

      \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B}\]
   5. Using strategy rm

   6. Applied associate-*r/ 19.3

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

   7. Taylor expanded around inf 0.1

      \[\leadsto \frac{\color{blue}{1 - 1 \cdot \frac{1}{{F}^{2}}}}{\sin B} - \frac{x \cdot 1}{\tan B}\]

   8. Simplified 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -7.39940744743415903 \cdot 10^{52}:\\ \;\;\;\;\frac{\frac{\frac{1}{F}}{F} - 1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 244697.137374221085:\\ \;\;\;\;\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{1}{\frac{\tan B}{x \cdot 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))