VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 0.2

Time: 11.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -282918228819822380000:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{elif}\;F \le 120765522752324900:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sin B}{F}}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \end{array}\]

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 temp;
	if ((F <= -2.8291822881982238e+20)) {
		temp = (((1.0 * (1.0 / (sin(B) * pow(F, 2.0)))) - (1.0 / sin(B))) + -((x * 1.0) / tan(B)));
	} else {
		double temp_1;
		if ((F <= 1.207655227523249e+17)) {
			temp_1 = fma((1.0 / (sin(B) / F)), pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)), -((x * 1.0) / tan(B)));
		} else {
			temp_1 = (((1.0 / sin(B)) - (1.0 * (1.0 / (sin(B) * pow(F, 2.0))))) + -((x * 1.0) / tan(B)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -2.8291822881982238e+20

   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 25.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)}\]
   3. Using strategy rm

   4. Applied associate-*r/ 25.2

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\color{blue}{\frac{x \cdot 1}{\tan B}}\right)\]
   5. Using strategy rm

   6. Applied clear-num 25.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\sin B}{F}}}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\frac{x \cdot 1}{\tan B}\right)\]
   7. Using strategy rm

   8. Applied fma-udef 25.2

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

   9. Simplified 19.5

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

   10. Taylor expanded around -inf 0.1

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

   if -2.8291822881982238e+20 < F < 1.207655227523249e+17

   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 0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)}\]
   3. Using strategy rm

   4. Applied associate-*r/ 0.3

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\color{blue}{\frac{x \cdot 1}{\tan B}}\right)\]
   5. Using strategy rm

   6. Applied clear-num 0.4

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

   if 1.207655227523249e+17 < F

   1. Initial program 26.9

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

   2. Simplified 26.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)}\]
   3. Using strategy rm

   4. Applied associate-*r/ 26.8

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\color{blue}{\frac{x \cdot 1}{\tan B}}\right)\]
   5. Using strategy rm

   6. Applied clear-num 26.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\sin B}{F}}}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\frac{x \cdot 1}{\tan B}\right)\]
   7. Using strategy rm

   8. Applied fma-udef 26.8

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

   9. Simplified 21.3

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

   10. Taylor expanded around inf 0.2

       \[\leadsto \color{blue}{\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)} + \left(-\frac{x \cdot 1}{\tan B}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -282918228819822380000:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{elif}\;F \le 120765522752324900:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sin B}{F}}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))