VandenBroeck and Keller, Equation (23)

Average Error: 14.0 → 0.2

Time: 35.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;F \le -4558293311076000858112:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \mathbf{elif}\;F \le 2.807063357183112132088549252493159419866 \cdot 10^{51}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

C

double f(double F, double B, double x) {
        double r78998 = x;
        double r78999 = 1.0;
        double r79000 = B;
        double r79001 = tan(r79000);
        double r79002 = r78999 / r79001;
        double r79003 = r78998 * r79002;
        double r79004 = -r79003;
        double r79005 = F;
        double r79006 = sin(r79000);
        double r79007 = r79005 / r79006;
        double r79008 = r79005 * r79005;
        double r79009 = 2.0;
        double r79010 = r79008 + r79009;
        double r79011 = r79009 * r78998;
        double r79012 = r79010 + r79011;
        double r79013 = r78999 / r79009;
        double r79014 = -r79013;
        double r79015 = pow(r79012, r79014);
        double r79016 = r79007 * r79015;
        double r79017 = r79004 + r79016;
        return r79017;
}

↓

double f(double F, double B, double x) {
        double r79018 = F;
        double r79019 = -4.558293311076001e+21;
        bool r79020 = r79018 <= r79019;
        double r79021 = x;
        double r79022 = 1.0;
        double r79023 = r79021 * r79022;
        double r79024 = B;
        double r79025 = tan(r79024);
        double r79026 = r79023 / r79025;
        double r79027 = -r79026;
        double r79028 = -1.0;
        double r79029 = sin(r79024);
        double r79030 = r79028 / r79029;
        double r79031 = 2.0;
        double r79032 = pow(r79018, r79031);
        double r79033 = r79029 * r79032;
        double r79034 = r79022 / r79033;
        double r79035 = r79030 + r79034;
        double r79036 = r79027 + r79035;
        double r79037 = 2.807063357183112e+51;
        bool r79038 = r79018 <= r79037;
        double r79039 = 2.0;
        double r79040 = fma(r79018, r79018, r79039);
        double r79041 = fma(r79039, r79021, r79040);
        double r79042 = r79022 / r79039;
        double r79043 = pow(r79041, r79042);
        double r79044 = r79043 * r79029;
        double r79045 = r79018 / r79044;
        double r79046 = r79027 + r79045;
        double r79047 = 1.0;
        double r79048 = r79047 / r79029;
        double r79049 = r79048 - r79034;
        double r79050 = r79027 + r79049;
        double r79051 = r79038 ? r79046 : r79050;
        double r79052 = r79020 ? r79036 : r79051;
        return r79052;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -4.558293311076001e+21

   1. Initial program 26.9

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

   3. Applied pow-neg 26.9

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

   4. Applied frac-times 20.8

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

   5. Simplified 20.8

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

   6. Simplified 20.8

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

   8. Applied associate-*r/ 20.7

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

   9. Taylor expanded around -inf 0.1

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

   10. Simplified 0.1

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

   if -4.558293311076001e+21 < F < 2.807063357183112e+51

   1. Initial program 0.6

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

   3. Applied pow-neg 0.6

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

   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

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

   if 2.807063357183112e+51 < F

   1. Initial program 29.0

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

   3. Applied pow-neg 29.0

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

   4. Applied frac-times 22.9

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

   5. Simplified 22.9

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

   6. Simplified 22.9

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

   8. Applied associate-*r/ 22.9

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

   9. Taylor expanded around inf 0.1

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

   10. Simplified 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \le -4558293311076000858112:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{-1}{\sin B} + \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \mathbf{elif}\;F \le 2.807063357183112132088549252493159419866 \cdot 10^{51}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{1}{2}\right)} \cdot \sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \left(\frac{1}{\sin B} - \frac{1}{\sin B \cdot {F}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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))))))