Average Error: 13.3 → 0.3
Time: 12.8s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -332253304756638400:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 191.051664676912566:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array}\]
\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 -332253304756638400:\\
\;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}\\

\mathbf{elif}\;F \le 191.051664676912566:\\
\;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\

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

\end{array}
double f(double F, double B, double x) {
        double r41192 = x;
        double r41193 = 1.0;
        double r41194 = B;
        double r41195 = tan(r41194);
        double r41196 = r41193 / r41195;
        double r41197 = r41192 * r41196;
        double r41198 = -r41197;
        double r41199 = F;
        double r41200 = sin(r41194);
        double r41201 = r41199 / r41200;
        double r41202 = r41199 * r41199;
        double r41203 = 2.0;
        double r41204 = r41202 + r41203;
        double r41205 = r41203 * r41192;
        double r41206 = r41204 + r41205;
        double r41207 = r41193 / r41203;
        double r41208 = -r41207;
        double r41209 = pow(r41206, r41208);
        double r41210 = r41201 * r41209;
        double r41211 = r41198 + r41210;
        return r41211;
}
↓
double f(double F, double B, double x) {
        double r41212 = F;
        double r41213 = -3.322533047566384e+17;
        bool r41214 = r41212 <= r41213;
        double r41215 = 1.0;
        double r41216 = 1.0;
        double r41217 = B;
        double r41218 = sin(r41217);
        double r41219 = 2.0;
        double r41220 = pow(r41212, r41219);
        double r41221 = r41218 * r41220;
        double r41222 = r41216 / r41221;
        double r41223 = r41215 * r41222;
        double r41224 = r41216 / r41218;
        double r41225 = r41223 - r41224;
        double r41226 = x;
        double r41227 = tan(r41217);
        double r41228 = r41215 / r41227;
        double r41229 = r41226 * r41228;
        double r41230 = r41225 - r41229;
        double r41231 = 191.05166467691257;
        bool r41232 = r41212 <= r41231;
        double r41233 = r41212 / r41218;
        double r41234 = r41212 * r41212;
        double r41235 = 2.0;
        double r41236 = r41234 + r41235;
        double r41237 = r41235 * r41226;
        double r41238 = r41236 + r41237;
        double r41239 = r41215 / r41235;
        double r41240 = pow(r41238, r41239);
        double r41241 = r41233 / r41240;
        double r41242 = r41226 * r41215;
        double r41243 = r41242 / r41227;
        double r41244 = r41241 - r41243;
        double r41245 = r41224 - r41223;
        double r41246 = r41245 - r41229;
        double r41247 = r41232 ? r41244 : r41246;
        double r41248 = r41214 ? r41230 : r41247;
        return r41248;
}

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 < -3.322533047566384e+17

    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. Simplified25.1

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

    3. Taylor expanded around -inf 0.2

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

    if -3.322533047566384e+17 < F < 191.05166467691257

    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. Simplified0.4

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

    4. Applied pow-neg0.4

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

    5. Applied un-div-inv0.4

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

    7. Applied associate-*r/0.3

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

    if 191.05166467691257 < F

    1. Initial program 24.0

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

    2. Simplified24.0

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

    3. Taylor expanded around inf 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -332253304756638400:\\ \;\;\;\;\left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right) - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \le 191.051664676912566:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{1}{2}\right)}} - \frac{x \cdot 1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right) - x \cdot \frac{1}{\tan B}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))))