Average Error: 0.0 → 0.0
Time: 22.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{1 - v \cdot v}{\frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \cdot \frac{4}{\sqrt{2}}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{1 - v \cdot v}{\frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \cdot \frac{4}{\sqrt{2}}}
double f(double v) {
        double r3508315 = 2.0;
        double r3508316 = sqrt(r3508315);
        double r3508317 = 4.0;
        double r3508318 = r3508316 / r3508317;
        double r3508319 = 1.0;
        double r3508320 = 3.0;
        double r3508321 = v;
        double r3508322 = r3508321 * r3508321;
        double r3508323 = r3508320 * r3508322;
        double r3508324 = r3508319 - r3508323;
        double r3508325 = sqrt(r3508324);
        double r3508326 = r3508318 * r3508325;
        double r3508327 = r3508319 - r3508322;
        double r3508328 = r3508326 * r3508327;
        return r3508328;
}
↓
double f(double v) {
        double r3508329 = 1.0;
        double r3508330 = v;
        double r3508331 = r3508330 * r3508330;
        double r3508332 = r3508329 - r3508331;
        double r3508333 = -3.0;
        double r3508334 = fma(r3508331, r3508333, r3508329);
        double r3508335 = sqrt(r3508334);
        double r3508336 = r3508329 / r3508335;
        double r3508337 = 4.0;
        double r3508338 = 2.0;
        double r3508339 = sqrt(r3508338);
        double r3508340 = r3508337 / r3508339;
        double r3508341 = r3508336 * r3508340;
        double r3508342 = r3508332 / r3508341;
        return r3508342;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{1 - v \cdot v}{\frac{\frac{4}{\sqrt{2}}}{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}}}\]
  3. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \frac{1 - v \cdot v}{\color{blue}{\frac{4}{\sqrt{2}} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}}}}\]

  5. Final simplification0.0

    \[\leadsto \frac{1 - v \cdot v}{\frac{1}{\sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \cdot \frac{4}{\sqrt{2}}}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))