VandenBroeck and Keller, Equation (23)

Average Error: 13.8 → 0.2

Time: 15.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \leq -18621346198.865578:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 111983907.8550106:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

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 -18621346198.865578)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 111983907.8550106)
     (- (/ (/ F (sin B)) (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) (/ x (tan B)))
     (- (/ 1.0 (sin B)) (/ 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 <= -18621346198.865578) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 111983907.8550106) {
		tmp = ((F / sin(B)) / sqrt((F * F) + (2.0 + (x * 2.0)))) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (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 < -18621346198.865578

   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

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

   3. Taylor expanded around -inf 0.2

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

   if -18621346198.865578 < F < 111983907.8550106

   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 add-sqr-sqrt_binary64 0.3

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

   5. Applied unpow-prod-down_binary64 0.3

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

   6. Simplified 0.3

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

   7. Simplified 0.3

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

   9. Applied pow1_binary64 0.3

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

   10. Applied pow1_binary64 0.3

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

   11. Applied pow-prod-down_binary64 0.3

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

   12. Simplified 0.3

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

   if 111983907.8550106 < F

   1. Initial program 25.3

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

   2. Simplified 25.2

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

   3. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -18621346198.865578:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 111983907.8550106:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))))))