VandenBroeck and Keller, Equation (23)

Average Error: 13.5 → 0.2

Time: 40.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -99469.4573886484722606837749481201171875:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \mathbf{elif}\;F \le 13711.08235068509384291246533393859863281:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\sin B} \cdot \cos B\right) + \frac{F}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

C

double f(double F, double B, double x) {
        double r103523 = x;
        double r103524 = 1.0;
        double r103525 = B;
        double r103526 = tan(r103525);
        double r103527 = r103524 / r103526;
        double r103528 = r103523 * r103527;
        double r103529 = -r103528;
        double r103530 = F;
        double r103531 = sin(r103525);
        double r103532 = r103530 / r103531;
        double r103533 = r103530 * r103530;
        double r103534 = 2.0;
        double r103535 = r103533 + r103534;
        double r103536 = r103534 * r103523;
        double r103537 = r103535 + r103536;
        double r103538 = r103524 / r103534;
        double r103539 = -r103538;
        double r103540 = pow(r103537, r103539);
        double r103541 = r103532 * r103540;
        double r103542 = r103529 + r103541;
        return r103542;
}

↓

double f(double F, double B, double x) {
        double r103543 = F;
        double r103544 = -99469.45738864847;
        bool r103545 = r103543 <= r103544;
        double r103546 = x;
        double r103547 = 1.0;
        double r103548 = r103546 * r103547;
        double r103549 = B;
        double r103550 = tan(r103549);
        double r103551 = r103548 / r103550;
        double r103552 = -r103551;
        double r103553 = -1.0;
        double r103554 = sin(r103549);
        double r103555 = r103553 / r103554;
        double r103556 = 2.0;
        double r103557 = pow(r103543, r103556);
        double r103558 = r103554 * r103557;
        double r103559 = r103547 / r103558;
        double r103560 = r103555 + r103559;
        double r103561 = r103552 + r103560;
        double r103562 = 13711.082350685094;
        bool r103563 = r103543 <= r103562;
        double r103564 = r103548 / r103554;
        double r103565 = cos(r103549);
        double r103566 = r103564 * r103565;
        double r103567 = -r103566;
        double r103568 = 2.0;
        double r103569 = fma(r103543, r103543, r103568);
        double r103570 = fma(r103568, r103546, r103569);
        double r103571 = r103547 / r103568;
        double r103572 = pow(r103570, r103571);
        double r103573 = r103572 * r103554;
        double r103574 = r103543 / r103573;
        double r103575 = r103567 + r103574;
        double r103576 = 1.0;
        double r103577 = r103576 / r103554;
        double r103578 = r103577 - r103559;
        double r103579 = r103552 + r103578;
        double r103580 = r103563 ? r103575 : r103579;
        double r103581 = r103545 ? r103561 : r103580;
        return r103581;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -99469.45738864847

   1. Initial program 24.2

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

   3. Applied pow-neg 24.2

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

   4. Applied frac-times 18.5

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

   5. Simplified 18.5

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

   6. Simplified 18.5

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

   8. Applied associate-*r/ 18.5

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

   9. Taylor expanded around -inf 0.2

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

   10. Simplified 0.2

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

   if -99469.45738864847 < F < 13711.082350685094

   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. Using strategy rm

   3. Applied pow-neg 0.4

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

   4. Applied frac-times 0.4

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

   5. Simplified 0.4

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

   6. Simplified 0.4

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

   8. Applied associate-*r/ 0.3

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

   10. Applied tan-quot 0.3

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

   11. Applied associate-/r/ 0.3

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

   if 13711.082350685094 < F

   1. Initial program 25.3

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

   3. Applied pow-neg 25.3

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

   4. Applied frac-times 19.8

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

   5. Simplified 19.8

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

   6. Simplified 19.8

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

   8. Applied associate-*r/ 19.7

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

   9. Taylor expanded around inf 0.2

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

   10. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -99469.4573886484722606837749481201171875:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \mathbf{elif}\;F \le 13711.08235068509384291246533393859863281:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\sin B} \cdot \cos B\right) + \frac{F}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))))