VandenBroeck and Keller, Equation (23)

Average Error: 13.8 → 0.2

Time: 16.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -41190.77908363891:\\ \;\;\;\;\begin{array}{l} t_2 := \sin B \cdot {F}^{2}\\ \left(\left(\frac{x}{t_2} + \frac{1}{t_2}\right) - t_0\right) - t_1 \end{array}\\ \mathbf{elif}\;F \leq 586476706.4778745:\\ \;\;\;\;\left(F \cdot \left(t_0 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right)\right) \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \]

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 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -41190.77908363891)
     (let* ((t_2 (* (sin B) (pow F 2.0))))
       (- (- (+ (/ x t_2) (/ 1.0 t_2)) t_0) t_1))
     (if (<= F 586476706.4778745)
       (-
        (*
         (* F (* t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.25)))
         (sqrt (pow (fma x 2.0 (fma F F 2.0)) -0.5)))
        t_1)
       (- t_0 t_1)))))

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 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -41190.77908363891) {
		double t_2_1 = sin(B) * pow(F, 2.0);
		tmp = (((x / t_2_1) + (1.0 / t_2_1)) - t_0) - t_1;
	} else if (F <= 586476706.4778745) {
		tmp = ((F * (t_0 * pow(fma(2.0, x, fma(F, F, 2.0)), -0.25))) * sqrt(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	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 < -41190.7790836389104

   1. Initial program 26.4

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

   2. Simplified 26.4

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

   3. Applied add-sqr-sqrt_binary64 26.4

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

   4. Applied associate-*r*_binary64 26.4

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

   5. Simplified 26.4

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

   6. Taylor expanded in F around -inf 0.2

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

   if -41190.7790836389104 < F < 586476706.47787452

   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

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

   3. Applied add-sqr-sqrt_binary64 0.6

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

   4. Applied associate-*r*_binary64 0.5

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

   5. Simplified 0.3

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

   6. Applied div-inv_binary64 0.3

      \[\leadsto \left(\color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right) \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - \frac{x}{\tan B} \]

   7. Applied associate-*l*_binary64 0.3

      \[\leadsto \color{blue}{\left(F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right)\right)} \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - \frac{x}{\tan B} \]

   if 586476706.47787452 < F

   1. Initial program 24.9

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

   2. Simplified 24.9

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

   3. Taylor expanded in F around inf 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -41190.77908363891:\\ \;\;\;\;\left(\left(\frac{x}{\sin B \cdot {F}^{2}} + \frac{1}{\sin B \cdot {F}^{2}}\right) - \frac{1}{\sin B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 586476706.4778745:\\ \;\;\;\;\left(F \cdot \left(\frac{1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right)\right) \cdot \sqrt{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))))