VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 0.2

Time: 14.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.1646893023852704 \cdot 10^{+55}:\\ \;\;\;\;\frac{F}{\left(\left(\frac{-1}{F} - F\right) - \frac{x}{F}\right) \cdot \sin B} - t_0\\ \mathbf{elif}\;F \leq 201452204.07493982:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - t_0\\ \end{array} \]

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
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.1646893023852704e+55)
     (- (/ F (* (- (- (/ -1.0 F) F) (/ x F)) (sin B))) t_0)
     (if (<= F 201452204.07493982)
       (- (/ (/ F (sqrt (+ 2.0 (+ (* F F) (* x 2.0))))) (sin B)) t_0)
       (- (/ (- 1.0 (+ (/ x (* F F)) (/ 1.0 (* F F)))) (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 t_0 = x / tan(B);
	double tmp;
	if (F <= -2.1646893023852704e+55) {
		tmp = (F / ((((-1.0 / F) - F) - (x / F)) * sin(B))) - t_0;
	} else if (F <= 201452204.07493982) {
		tmp = ((F / sqrt(2.0 + ((F * F) + (x * 2.0)))) / sin(B)) - t_0;
	} else {
		tmp = ((1.0 - ((x / (F * F)) + (1.0 / (F * F)))) / sin(B)) - t_0;
	}
	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 < -2.16468930238527039e55

   1. Initial program 28.7

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

   2. Simplified 28.6

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}} \]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 22.1

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

   5. Simplified 22.1

      \[\leadsto \frac{\color{blue}{F \cdot {\left(2 + \left(2 \cdot x + F \cdot F\right)\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]
   6. Using strategy rm

   7. Applied associate-/l*_binary64 22.1

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

   8. Simplified 22.1

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

   9. Taylor expanded around inf 22.1

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

   10. Simplified 22.1

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

   11. Taylor expanded around -inf 0.2

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

   12. Simplified 0.2

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

   if -2.16468930238527039e55 < F < 201452204.074939817

   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

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}} \]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 0.3

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

   5. Simplified 0.3

      \[\leadsto \frac{\color{blue}{F \cdot {\left(2 + \left(2 \cdot x + F \cdot F\right)\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]
   6. Using strategy rm

   7. Applied associate-/l*_binary64 0.3

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

   8. Simplified 0.3

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

   9. Taylor expanded around inf 0.3

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

   10. Simplified 0.3

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

   12. Applied associate-/r*_binary64 0.3

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

   13. Simplified 0.3

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

   if 201452204.074939817 < 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

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}} \]
   3. Using strategy rm

   4. Applied associate-*l/_binary64 19.9

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

   5. Simplified 19.9

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

   6. Taylor expanded around inf 0.2

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

   7. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1646893023852704 \cdot 10^{+55}:\\ \;\;\;\;\frac{F}{\left(\left(\frac{-1}{F} - F\right) - \frac{x}{F}\right) \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 201452204.07493982:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \left(F \cdot F + x \cdot 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{x}{F \cdot F} + \frac{1}{F \cdot F}\right)}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))))