Average Error: 14.1 → 11.1
Time: 33.7s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot F\right)}{\sin B} - \frac{x}{\tan B}\]
\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)}
↓
\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot F\right)}{\sin B} - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r2039507 = x;
        double r2039508 = 1.0;
        double r2039509 = B;
        double r2039510 = tan(r2039509);
        double r2039511 = r2039508 / r2039510;
        double r2039512 = r2039507 * r2039511;
        double r2039513 = -r2039512;
        double r2039514 = F;
        double r2039515 = sin(r2039509);
        double r2039516 = r2039514 / r2039515;
        double r2039517 = r2039514 * r2039514;
        double r2039518 = 2.0;
        double r2039519 = r2039517 + r2039518;
        double r2039520 = r2039518 * r2039507;
        double r2039521 = r2039519 + r2039520;
        double r2039522 = r2039508 / r2039518;
        double r2039523 = -r2039522;
        double r2039524 = pow(r2039521, r2039523);
        double r2039525 = r2039516 * r2039524;
        double r2039526 = r2039513 + r2039525;
        return r2039526;
}
↓
double f(double F, double B, double x) {
        double r2039527 = 2.0;
        double r2039528 = x;
        double r2039529 = F;
        double r2039530 = fma(r2039529, r2039529, r2039527);
        double r2039531 = fma(r2039527, r2039528, r2039530);
        double r2039532 = -0.25;
        double r2039533 = pow(r2039531, r2039532);
        double r2039534 = r2039533 * r2039529;
        double r2039535 = r2039533 * r2039534;
        double r2039536 = B;
        double r2039537 = sin(r2039536);
        double r2039538 = r2039535 / r2039537;
        double r2039539 = tan(r2039536);
        double r2039540 = r2039528 / r2039539;
        double r2039541 = r2039538 - r2039540;
        return r2039541;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 14.1

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

  2. Simplified14.0

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

  4. Applied associate-*r/11.0

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

  6. Applied sqr-pow11.0

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

  7. Applied associate-*l*11.1

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

  8. Final simplification11.1

    \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{4}} \cdot F\right)}{\sin B} - \frac{x}{\tan B}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  (+ (- (* x (/ 1 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2) (* 2 x)) (- (/ 1 2))))))