Average Error: 13.7 → 10.8
Time: 40.3s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \left(\mathsf{fma}\left(F, F, 2\right)\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}\]
\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{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \left(\mathsf{fma}\left(F, F, 2\right)\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r1541000 = x;
        double r1541001 = 1.0;
        double r1541002 = B;
        double r1541003 = tan(r1541002);
        double r1541004 = r1541001 / r1541003;
        double r1541005 = r1541000 * r1541004;
        double r1541006 = -r1541005;
        double r1541007 = F;
        double r1541008 = sin(r1541002);
        double r1541009 = r1541007 / r1541008;
        double r1541010 = r1541007 * r1541007;
        double r1541011 = 2.0;
        double r1541012 = r1541010 + r1541011;
        double r1541013 = r1541011 * r1541000;
        double r1541014 = r1541012 + r1541013;
        double r1541015 = r1541001 / r1541011;
        double r1541016 = -r1541015;
        double r1541017 = pow(r1541014, r1541016);
        double r1541018 = r1541009 * r1541017;
        double r1541019 = r1541006 + r1541018;
        return r1541019;
}
↓
double f(double F, double B, double x) {
        double r1541020 = 1.0;
        double r1541021 = B;
        double r1541022 = sin(r1541021);
        double r1541023 = r1541020 / r1541022;
        double r1541024 = 2.0;
        double r1541025 = x;
        double r1541026 = F;
        double r1541027 = fma(r1541026, r1541026, r1541024);
        double r1541028 = fma(r1541024, r1541025, r1541027);
        double r1541029 = -0.5;
        double r1541030 = pow(r1541028, r1541029);
        double r1541031 = r1541030 * r1541026;
        double r1541032 = r1541023 * r1541031;
        double r1541033 = tan(r1541021);
        double r1541034 = r1541025 / r1541033;
        double r1541035 = r1541032 - r1541034;
        return r1541035;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.7

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

  2. Simplified13.7

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

  4. Applied div-inv13.7

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

  5. Applied associate-*r*10.8

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

  6. Final simplification10.8

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

Reproduce

herbie shell --seed 2019130 +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))))))