VandenBroeck and Keller, Equation (23)

Average Error: 13.5 → 0.2

Time: 17.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -99771073.7748098224:\\ \;\;\;\;\left(-\frac{x}{\frac{\tan B}{1}}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 3.99929698532464133 \cdot 10^{93}:\\ \;\;\;\;\left(-\frac{x}{\frac{\tan B}{1}}\right) + \frac{1}{\left(-\sin B\right) \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}} \cdot \frac{-F}{{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{\frac{\tan B}{1}}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

C

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

↓

double code(double F, double B, double x) {
	double VAR;
	if ((F <= -99771073.77480982)) {
		VAR = ((double) (((double) -(((double) (x / ((double) (((double) tan(B)) / 1.0)))))) + ((double) (((double) (1.0 * ((double) (1.0 / ((double) (((double) sin(B)) * ((double) pow(F, 2.0)))))))) - ((double) (1.0 / ((double) sin(B))))))));
	} else {
		double VAR_1;
		if ((F <= 3.9992969853246413e+93)) {
			VAR_1 = ((double) (((double) -(((double) (x / ((double) (((double) tan(B)) / 1.0)))))) + ((double) (((double) (1.0 / ((double) (((double) -(((double) sin(B)))) * ((double) pow(((double) sqrt(((double) (((double) (((double) (F * F)) + 2.0)) + ((double) (2.0 * x)))))), ((double) (1.0 / 2.0)))))))) * ((double) (((double) -(F)) / ((double) pow(((double) sqrt(((double) (((double) (((double) (F * F)) + 2.0)) + ((double) (2.0 * x)))))), ((double) (1.0 / 2.0))))))))));
		} else {
			VAR_1 = ((double) (((double) -(((double) (x / ((double) (((double) tan(B)) / 1.0)))))) + ((double) (((double) (1.0 / ((double) sin(B)))) - ((double) (1.0 * ((double) (1.0 / ((double) (((double) sin(B)) * ((double) pow(F, 2.0))))))))))));
		}
		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 < -99771073.77480982

   1. Initial program 25.6

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

   3. Applied pow-neg 25.6

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

   4. Applied frac-2neg 25.6

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

   5. Applied frac-times 19.4

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

   6. Simplified 19.4

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

   8. Applied clear-num 19.4

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

   9. Applied un-div-inv 19.4

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

   10. Taylor expanded around -inf 0.1

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

   if -99771073.77480982 < F < 3.9992969853246413e+93

   1. Initial program 0.8

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

   3. Applied pow-neg 0.9

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

   4. Applied frac-2neg 0.9

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

   5. Applied frac-times 0.4

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

   6. Simplified 0.4

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

   8. Applied clear-num 0.4

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

   9. Applied un-div-inv 0.3

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

   11. Applied add-sqr-sqrt 0.3

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

   12. Applied unpow-prod-down 0.3

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

   13. Applied associate-*r* 0.3

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

   14. Applied *-un-lft-identity 0.3

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

   15. Applied times-frac 0.3

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

   if 3.9992969853246413e+93 < F

   1. Initial program 31.9

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

   3. Applied pow-neg 31.9

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

   4. Applied frac-2neg 31.9

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

   5. Applied frac-times 25.6

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

   6. Simplified 25.6

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

   8. Applied clear-num 25.6

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

   9. Applied un-div-inv 25.5

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

   10. Taylor expanded around inf 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -99771073.7748098224:\\ \;\;\;\;\left(-\frac{x}{\frac{\tan B}{1}}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 3.99929698532464133 \cdot 10^{93}:\\ \;\;\;\;\left(-\frac{x}{\frac{\tan B}{1}}\right) + \frac{1}{\left(-\sin B\right) \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}} \cdot \frac{-F}{{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(\frac{1}{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{\frac{\tan B}{1}}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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))))))