Average Error: 13.3 → 10.2
Time: 2.5m
Precision: 64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\]
↓
\[\left({\left(\mathsf{fma}\left(x, 2, \left(\mathsf{fma}\left(F, F, 2\right)\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\sin B}\right) \cdot F - \frac{x}{\tan B}\]
\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)}
↓
\left({\left(\mathsf{fma}\left(x, 2, \left(\mathsf{fma}\left(F, F, 2\right)\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{1}{\sin B}\right) \cdot F - \frac{x}{\tan B}
double f(double F, double B, double x) {
        double r9096398 = x;
        double r9096399 = 1.0;
        double r9096400 = B;
        double r9096401 = tan(r9096400);
        double r9096402 = r9096399 / r9096401;
        double r9096403 = r9096398 * r9096402;
        double r9096404 = -r9096403;
        double r9096405 = F;
        double r9096406 = sin(r9096400);
        double r9096407 = r9096405 / r9096406;
        double r9096408 = r9096405 * r9096405;
        double r9096409 = 2.0;
        double r9096410 = r9096408 + r9096409;
        double r9096411 = r9096409 * r9096398;
        double r9096412 = r9096410 + r9096411;
        double r9096413 = r9096399 / r9096409;
        double r9096414 = -r9096413;
        double r9096415 = pow(r9096412, r9096414);
        double r9096416 = r9096407 * r9096415;
        double r9096417 = r9096404 + r9096416;
        return r9096417;
}
↓
double f(double F, double B, double x) {
        double r9096418 = x;
        double r9096419 = 2.0;
        double r9096420 = F;
        double r9096421 = fma(r9096420, r9096420, r9096419);
        double r9096422 = fma(r9096418, r9096419, r9096421);
        double r9096423 = -0.5;
        double r9096424 = pow(r9096422, r9096423);
        double r9096425 = 1.0;
        double r9096426 = B;
        double r9096427 = sin(r9096426);
        double r9096428 = r9096425 / r9096427;
        double r9096429 = r9096424 * r9096428;
        double r9096430 = r9096429 * r9096420;
        double r9096431 = tan(r9096426);
        double r9096432 = r9096418 / r9096431;
        double r9096433 = r9096430 - r9096432;
        return r9096433;
}

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

  1. Initial program 13.3

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

  2. Simplified12.7

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

  4. Applied div-inv12.8

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

  5. Applied *-un-lft-identity12.8

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

  6. Applied unpow-prod-down12.8

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

  7. Applied times-frac10.2

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

  8. Simplified10.2

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

  9. Simplified10.2

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

  11. Applied associate-*r*10.2

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

  12. Final simplification10.2

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

Reproduce

herbie shell --seed 2019121 +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))))))