VandenBroeck and Keller, Equation (23)

Average Error: 13.6 → 0.2

Time: 9.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -158492.60646815697:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 192622.273679248086:\\ \;\;\;\;\frac{\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

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 <= -158492.60646815697)) {
		VAR = (((1.0 * (1.0 / (sin(B) * pow(F, 2.0)))) - (1.0 / sin(B))) - ((x * 1.0) / tan(B)));
	} else {
		double VAR_1;
		if ((F <= 192622.2736792481)) {
			VAR_1 = (((F / (sin(B) * pow((((F * F) + 2.0) + (2.0 * x)), ((1.0 / 2.0) / 2.0)))) / pow((((F * F) + 2.0) + (2.0 * x)), ((1.0 / 2.0) / 2.0))) - ((x * 1.0) / tan(B)));
		} else {
			VAR_1 = (((1.0 / sin(B)) - (1.0 * (1.0 / (sin(B) * pow(F, 2.0))))) - ((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 < -158492.60646815697

   1. Initial program 24.8

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

   2. Simplified 24.8

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}}\]
   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/ 19.4

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

   7. Taylor expanded around -inf 0.1

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

   if -158492.60646815697 < F < 192622.2736792481

   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 0.4

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}}\]
   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 pow-neg 0.3

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

   9. Applied associate-*r/ 0.3

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

   10. Applied associate-/l/ 0.3

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

   12. Applied sqr-pow 0.3

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

   13. Applied associate-*r* 0.3

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

   14. Applied associate-/r* 0.3

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

   15. Simplified 0.3

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

   if 192622.2736792481 < 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

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

   4. Applied associate-*l/ 20.4

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

   6. Applied associate-*r/ 20.4

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

   8. Applied pow-neg 20.4

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

   9. Applied associate-*r/ 20.4

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

   10. Applied associate-/l/ 20.4

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

   11. Taylor expanded around inf 0.2

       \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)} - \frac{x \cdot 1}{\tan B}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -158492.60646815697:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - \frac{x \cdot 1}{\tan B}\\ \mathbf{elif}\;F \le 192622.273679248086:\\ \;\;\;\;\frac{\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{x \cdot 1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(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))))))