Average Error: 14.0 → 0.3
Time: 11.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1.03118247158206771 \cdot 10^{43}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\ \mathbf{elif}\;F \le 4.0460331114730965 \cdot 10^{65}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) + \left(-x \cdot \frac{1}{\tan B}\right)\\ \end{array}\]
\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 -1.03118247158206771 \cdot 10^{43}:\\
\;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\

\mathbf{elif}\;F \le 4.0460331114730965 \cdot 10^{65}:\\
\;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) + \left(-x \cdot \frac{1}{\tan B}\right)\\

\end{array}
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 <= -1.0311824715820677e+43)) {
		VAR = fma(1.0, (x / (sin(B) * pow(F, 2.0))), -fma(1.0, ((x * cos(B)) / sin(B)), (1.0 / sin(B))));
	} else {
		double VAR_1;
		if ((F <= 4.0460331114730965e+65)) {
			VAR_1 = ((F / (sin(B) * pow((((F * F) + 2.0) + (2.0 * x)), (1.0 / 2.0)))) + -(x * (1.0 / tan(B))));
		} else {
			VAR_1 = (((1.0 / sin(B)) - (1.0 * (1.0 / (sin(B) * pow(F, 2.0))))) + -(x * (1.0 / tan(B))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if F < -1.0311824715820677e+43

    1. Initial program 27.3

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

    2. Simplified27.3

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1.0311824715820677e+43 < F < 4.0460331114730965e+65

    1. Initial program 0.7

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

    2. Simplified0.7

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

    4. Applied pow-neg0.8

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

    6. Applied fma-udef0.8

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

    7. Simplified0.7

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

    9. Applied div-inv0.8

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

    10. Applied associate-/l*0.4

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

    11. Simplified0.4

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

    if 4.0460331114730965e+65 < F

    1. Initial program 29.9

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

    2. Simplified29.9

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

    4. Applied pow-neg29.9

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

    6. Applied fma-udef29.9

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

    7. Simplified29.9

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

    8. Taylor expanded around inf 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1.03118247158206771 \cdot 10^{43}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{\sin B \cdot {F}^{2}}, -\mathsf{fma}\left(1, \frac{x \cdot \cos B}{\sin B}, \frac{1}{\sin B}\right)\right)\\ \mathbf{elif}\;F \le 4.0460331114730965 \cdot 10^{65}:\\ \;\;\;\;\frac{F}{\sin B \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) + \left(-x \cdot \frac{1}{\tan B}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +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))))))