Average Error: 13.6 → 10.8
Time: 38.9s
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \left(x \cdot \cos B\right) \cdot \frac{1}{\sin B}\]
\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, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) - \left(x \cdot \cos B\right) \cdot \frac{1}{\sin B}
double f(double F, double B, double x) {
        double r1790175 = x;
        double r1790176 = 1.0;
        double r1790177 = B;
        double r1790178 = tan(r1790177);
        double r1790179 = r1790176 / r1790178;
        double r1790180 = r1790175 * r1790179;
        double r1790181 = -r1790180;
        double r1790182 = F;
        double r1790183 = sin(r1790177);
        double r1790184 = r1790182 / r1790183;
        double r1790185 = r1790182 * r1790182;
        double r1790186 = 2.0;
        double r1790187 = r1790185 + r1790186;
        double r1790188 = r1790186 * r1790175;
        double r1790189 = r1790187 + r1790188;
        double r1790190 = r1790176 / r1790186;
        double r1790191 = -r1790190;
        double r1790192 = pow(r1790189, r1790191);
        double r1790193 = r1790184 * r1790192;
        double r1790194 = r1790181 + r1790193;
        return r1790194;
}
↓
double f(double F, double B, double x) {
        double r1790195 = 1.0;
        double r1790196 = B;
        double r1790197 = sin(r1790196);
        double r1790198 = r1790195 / r1790197;
        double r1790199 = 2.0;
        double r1790200 = x;
        double r1790201 = F;
        double r1790202 = fma(r1790201, r1790201, r1790199);
        double r1790203 = fma(r1790199, r1790200, r1790202);
        double r1790204 = -0.5;
        double r1790205 = pow(r1790203, r1790204);
        double r1790206 = r1790205 * r1790201;
        double r1790207 = r1790198 * r1790206;
        double r1790208 = cos(r1790196);
        double r1790209 = r1790200 * r1790208;
        double r1790210 = r1790209 * r1790198;
        double r1790211 = r1790207 - r1790210;
        return r1790211;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.6

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

  2. Simplified13.5

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

  4. Applied div-inv13.5

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

  5. Applied associate-*r*10.7

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

  6. Taylor expanded around inf 10.7

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

  8. Applied div-inv10.8

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

  9. Final simplification10.8

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

Reproduce

herbie shell --seed 2019158 +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))))))