VandenBroeck and Keller, Equation (23)

Average Error: 14.0 → 0.3

Time: 11.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -8.50305803332365509 \cdot 10^{158}:\\ \;\;\;\;\frac{\frac{\frac{1}{F}}{F} - 1}{\sin B} + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{elif}\;F \le 50548945.160980105:\\ \;\;\;\;\frac{\frac{F}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}}}{\sin B} + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \end{array}\]

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 <= -8.503058033323655e+158)) {
		VAR = (((((1.0 / F) / F) - 1.0) / sin(B)) + -((x * 1.0) / tan(B)));
	} else {
		double VAR_1;
		if ((F <= 50548945.160980105)) {
			VAR_1 = (((F / pow(fma(F, F, fma(2.0, x, 2.0)), (1.0 / 2.0))) / sin(B)) + -((x * 1.0) / tan(B)));
		} 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 < -8.503058033323655e+158

   1. Initial program 41.4

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

   2. Simplified 41.4

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

   4. Applied sqr-pow 41.4

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

   5. Simplified 41.4

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

   6. Simplified 41.4

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

   8. Applied fma-udef 41.4

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

   9. Simplified 35.9

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

   11. Applied associate-*r/ 35.9

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

   12. Taylor expanded around -inf 0.1

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

   13. Simplified 0.1

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

   if -8.503058033323655e+158 < F < 50548945.160980105

   1. Initial program 2.0

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

   2. Simplified 2.0

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

   4. Applied sqr-pow 2.0

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

   5. Simplified 2.0

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

   6. Simplified 2.0

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

   8. Applied fma-udef 2.0

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

   9. Simplified 0.6

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

   11. Applied associate-*r/ 0.5

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

   13. Applied pow-neg 0.5

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

   14. Applied associate-*l/ 0.4

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

   15. Simplified 0.4

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

   if 50548945.160980105 < F

   1. Initial program 25.8

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

   2. Simplified 25.8

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

   4. Applied sqr-pow 25.9

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

   5. Simplified 25.9

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

   6. Simplified 25.8

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

   8. Applied fma-udef 25.8

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

   9. Simplified 20.1

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

   11. Applied associate-*r/ 20.1

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

   12. Taylor expanded around inf 0.1

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

   13. Simplified 0.1

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -8.50305803332365509 \cdot 10^{158}:\\ \;\;\;\;\frac{\frac{\frac{1}{F}}{F} - 1}{\sin B} + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{elif}\;F \le 50548945.160980105:\\ \;\;\;\;\frac{\frac{F}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)}}}{\sin B} + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{1}{F}}{F}}{\sin B} + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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))))))