VandenBroeck and Keller, Equation (23)

Average Error: 13.7 → 0.2

Time: 9.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -165734231225465.62:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 108858170.1752173:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B \cdot \left(1 + \frac{x}{F \cdot F}\right)} - \frac{x}{\tan B}\\ \end{array}\]

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
 (if (<= F -165734231225465.62)
   (- (/ (- (/ 1.0 (* F F)) 1.0) (sin B)) (/ x (tan B)))
   (if (<= F 108858170.1752173)
     (-
      (/ 1.0 (/ (sin B) (* F (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))))
      (/ x (tan B)))
     (- (/ 1.0 (* (sin B) (+ 1.0 (/ x (* F F))))) (/ x (tan B))))))

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 tmp;
	if (F <= -165734231225465.62) {
		tmp = (((1.0 / (F * F)) - 1.0) / sin(B)) - (x / tan(B));
	} else if (F <= 108858170.1752173) {
		tmp = (1.0 / (sin(B) / (F * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)))) - (x / tan(B));
	} else {
		tmp = (1.0 / (sin(B) * (1.0 + (x / (F * F))))) - (x / tan(B));
	}
	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 < -165734231225465.62

   1. Initial program 25.4

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

   2. Simplified 25.4

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

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

   6. Taylor expanded around -inf 0.1

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

   7. Simplified 0.1

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

   if -165734231225465.62 < F < 108858170.1752173

   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 0.3

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

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

   7. Applied clear-num_binary64 0.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}}} - \frac{x}{\tan B}\]

   if 108858170.1752173 < F

   1. Initial program 25.0

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

   2. Simplified 25.0

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

   4. Applied associate-*l/_binary64 19.3

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

   5. Simplified 19.3

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

   7. Applied clear-num_binary64 19.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5}}}} - \frac{x}{\tan B}\]

   8. Taylor expanded around inf 0.1

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

   9. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -165734231225465.62:\\ \;\;\;\;\frac{\frac{1}{F \cdot F} - 1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 108858170.1752173:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B \cdot \left(1 + \frac{x}{F \cdot F}\right)} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))))))