Average Error: 0.0 → 0.0
Time: 2.4m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}{\frac{4}{1 - v \cdot v}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)}}{\frac{4}{1 - v \cdot v}}
double f(double v) {
        double r34400300 = 2.0;
        double r34400301 = sqrt(r34400300);
        double r34400302 = 4.0;
        double r34400303 = r34400301 / r34400302;
        double r34400304 = 1.0;
        double r34400305 = 3.0;
        double r34400306 = v;
        double r34400307 = r34400306 * r34400306;
        double r34400308 = r34400305 * r34400307;
        double r34400309 = r34400304 - r34400308;
        double r34400310 = sqrt(r34400309);
        double r34400311 = r34400303 * r34400310;
        double r34400312 = r34400304 - r34400307;
        double r34400313 = r34400311 * r34400312;
        return r34400313;
}
↓
double f(double v) {
        double r34400314 = 2.0;
        double r34400315 = sqrt(r34400314);
        double r34400316 = v;
        double r34400317 = r34400316 * r34400316;
        double r34400318 = -3.0;
        double r34400319 = 1.0;
        double r34400320 = fma(r34400317, r34400318, r34400319);
        double r34400321 = sqrt(r34400320);
        double r34400322 = r34400315 * r34400321;
        double r34400323 = 4.0;
        double r34400324 = r34400319 - r34400317;
        double r34400325 = r34400323 / r34400324;
        double r34400326 = r34400322 / r34400325;
        return r34400326;
}

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

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

  4. Applied associate-*l/0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))