Average Error: 13.4 → 0.2
Time: 13.1s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;F \le -1354270687656807936:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 23947.60619441725793876685202121734619141:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]
\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 -1354270687656807936:\\
\;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\

\mathbf{elif}\;F \le 23947.60619441725793876685202121734619141:\\
\;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\

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

\end{array}
double f(double F, double B, double x) {
        double r42363 = x;
        double r42364 = 1.0;
        double r42365 = B;
        double r42366 = tan(r42365);
        double r42367 = r42364 / r42366;
        double r42368 = r42363 * r42367;
        double r42369 = -r42368;
        double r42370 = F;
        double r42371 = sin(r42365);
        double r42372 = r42370 / r42371;
        double r42373 = r42370 * r42370;
        double r42374 = 2.0;
        double r42375 = r42373 + r42374;
        double r42376 = r42374 * r42363;
        double r42377 = r42375 + r42376;
        double r42378 = r42364 / r42374;
        double r42379 = -r42378;
        double r42380 = pow(r42377, r42379);
        double r42381 = r42372 * r42380;
        double r42382 = r42369 + r42381;
        return r42382;
}
↓
double f(double F, double B, double x) {
        double r42383 = F;
        double r42384 = -1.354270687656808e+18;
        bool r42385 = r42383 <= r42384;
        double r42386 = x;
        double r42387 = 1.0;
        double r42388 = r42386 * r42387;
        double r42389 = B;
        double r42390 = tan(r42389);
        double r42391 = r42388 / r42390;
        double r42392 = -r42391;
        double r42393 = 1.0;
        double r42394 = sin(r42389);
        double r42395 = 2.0;
        double r42396 = pow(r42383, r42395);
        double r42397 = r42394 * r42396;
        double r42398 = r42393 / r42397;
        double r42399 = r42387 * r42398;
        double r42400 = r42393 / r42394;
        double r42401 = r42399 - r42400;
        double r42402 = r42392 + r42401;
        double r42403 = 23947.606194417258;
        bool r42404 = r42383 <= r42403;
        double r42405 = cos(r42389);
        double r42406 = r42386 * r42405;
        double r42407 = r42406 / r42394;
        double r42408 = r42387 * r42407;
        double r42409 = -r42408;
        double r42410 = r42383 / r42394;
        double r42411 = r42383 * r42383;
        double r42412 = 2.0;
        double r42413 = r42411 + r42412;
        double r42414 = r42412 * r42386;
        double r42415 = r42413 + r42414;
        double r42416 = r42387 / r42412;
        double r42417 = -r42416;
        double r42418 = pow(r42415, r42417);
        double r42419 = r42410 * r42418;
        double r42420 = r42409 + r42419;
        double r42421 = r42400 - r42399;
        double r42422 = r42392 + r42421;
        double r42423 = r42404 ? r42420 : r42422;
        double r42424 = r42385 ? r42402 : r42423;
        return r42424;
}

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 < -1.354270687656808e+18

    1. Initial program 24.5

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

    3. Applied pow-neg24.5

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

    4. Applied frac-times19.1

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

    5. Simplified19.1

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

    7. Applied associate-*r/19.0

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

    8. Taylor expanded around -inf 0.2

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

    if -1.354270687656808e+18 < F < 23947.606194417258

    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. Taylor expanded around inf 0.3

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

    if 23947.606194417258 < F

    1. Initial program 25.5

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

    3. Applied pow-neg25.5

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

    4. Applied frac-times19.4

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

    5. Simplified19.4

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

    7. Applied associate-*r/19.3

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

    8. Taylor expanded around inf 0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -1354270687656807936:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(1 \cdot \frac{1}{\sin B \cdot {F}^{2}} - \frac{1}{\sin B}\right)\\ \mathbf{elif}\;F \le 23947.60619441725793876685202121734619141:\\ \;\;\;\;\left(-1 \cdot \frac{x \cdot \cos B}{\sin B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - 1 \cdot \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +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))))))