VandenBroeck and Keller, Equation (23)

Average Error: 13.7 → 0.2

Time: 14.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \left(F \cdot F\right) \cdot \sin B\\ t_2 := \frac{x}{t_1} + \frac{1}{t_1}\\ \mathbf{if}\;F \leq -21717636853701800:\\ \;\;\;\;\left(t_2 + \frac{-1}{\sin B}\right) - t_0\\ \mathbf{elif}\;F \leq 80529.85004951496:\\ \;\;\;\;\left(\frac{F}{\sin B} \cdot {\left(2 + \left(F \cdot F + x \cdot 2\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}^{-0.5}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - t_2\right) - t_0\\ \end{array} \]

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)))
        (t_1 (* (* F F) (sin B)))
        (t_2 (+ (/ x t_1) (/ 1.0 t_1))))
   (if (<= F -21717636853701800.0)
     (- (+ t_2 (/ -1.0 (sin B))) t_0)
     (if (<= F 80529.85004951496)
       (-
        (*
         (* (/ F (sin B)) (pow (+ 2.0 (+ (* F F) (* x 2.0))) -0.25))
         (sqrt (pow (+ (* x 2.0) (+ (* F F) 2.0)) -0.5)))
        t_0)
       (- (- (/ 1.0 (sin B)) t_2) 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 t_1 = (F * F) * sin(B);
	double t_2 = (x / t_1) + (1.0 / t_1);
	double tmp;
	if (F <= -21717636853701800.0) {
		tmp = (t_2 + (-1.0 / sin(B))) - t_0;
	} else if (F <= 80529.85004951496) {
		tmp = (((F / sin(B)) * pow((2.0 + ((F * F) + (x * 2.0))), -0.25)) * sqrt(pow(((x * 2.0) + ((F * F) + 2.0)), -0.5))) - t_0;
	} else {
		tmp = ((1.0 / sin(B)) - t_2) - 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 < -21717636853701800

   1. Initial program 25.1

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

   2. Simplified 25.1

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

   3. Taylor expanded around -inf 0.1

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

   4. Simplified 0.1

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

   if -21717636853701800 < F < 80529.850049514964

   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 0.6

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

   5. Applied associate-*r*_binary64 0.5

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

   6. Simplified 0.3

      \[\leadsto \color{blue}{\left(\frac{F}{\sin B} \cdot {\left(2 + \left(2 \cdot x + F \cdot F\right)\right)}^{-0.25}\right)} \cdot \sqrt{{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}} - \frac{x}{\tan B} \]

   if 80529.850049514964 < 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. Taylor expanded around inf 0.2

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

   4. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -21717636853701800:\\ \;\;\;\;\left(\left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right) + \frac{-1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 80529.85004951496:\\ \;\;\;\;\left(\frac{F}{\sin B} \cdot {\left(2 + \left(F \cdot F + x \cdot 2\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}^{-0.5}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - \left(\frac{x}{\left(F \cdot F\right) \cdot \sin B} + \frac{1}{\left(F \cdot F\right) \cdot \sin B}\right)\right) - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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))))))