VandenBroeck and Keller, Equation (23)

Average Error: 14.1 → 0.3

Time: 8.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -7.89324922578734301 \cdot 10^{120}:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{elif}\;F \le 175693108.13025206:\\ \;\;\;\;\frac{1}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}} \cdot F + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\sin B}, \cos B, \frac{x}{\sin B \cdot {F}^{2}}\right), -1, \frac{1}{\sin B}\right)\\ \end{array}\]

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 VAR;
	if ((F <= -7.893249225787343e+120)) {
		VAR = (((1.0 * (1.0 / (sin(B) * pow(F, 2.0)))) - (1.0 / sin(B))) + -((x * 1.0) / tan(B)));
	} else {
		double VAR_1;
		if ((F <= 175693108.13025206)) {
			VAR_1 = (((1.0 / (sin(B) / pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))) * F) + -((x * 1.0) / tan(B)));
		} else {
			VAR_1 = fma(fma((x / sin(B)), cos(B), (x / (sin(B) * pow(F, 2.0)))), -1.0, (1.0 / sin(B)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -7.893249225787343e+120

   1. Initial program 36.5

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

   2. Simplified 36.5

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

   4. Applied associate-*r/ 36.5

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

   6. Applied clear-num 36.5

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

   8. Applied fma-udef 36.5

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

   9. Simplified 30.7

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

   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 -7.893249225787343e+120 < F < 175693108.13025206

   1. Initial program 1.2

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

   2. Simplified 1.2

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

   4. Applied associate-*r/ 1.1

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

   6. Applied clear-num 1.1

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

   8. Applied fma-udef 1.1

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

   9. Simplified 0.3

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

   11. Applied clear-num 0.3

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

   if 175693108.13025206 < F

   1. Initial program 26.3

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

   2. Simplified 26.3

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

   4. Applied associate-*r/ 26.3

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

   5. Taylor expanded around inf 0.2

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

   6. Simplified 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -7.89324922578734301 \cdot 10^{120}:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{elif}\;F \le 175693108.13025206:\\ \;\;\;\;\frac{1}{\frac{\sin B}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}} \cdot F + \left(-\frac{x \cdot 1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\sin B}, \cos B, \frac{x}{\sin B \cdot {F}^{2}}\right), -1, \frac{1}{\sin B}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))))))