Average Error: 0.0 → 0.0
Time: 17.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\mathsf{fma}\left(-v, v, 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\]
\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}}{4} \cdot \left(\mathsf{fma}\left(-v, v, 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)
double f(double v) {
        double r178934 = 2.0;
        double r178935 = sqrt(r178934);
        double r178936 = 4.0;
        double r178937 = r178935 / r178936;
        double r178938 = 1.0;
        double r178939 = 3.0;
        double r178940 = v;
        double r178941 = r178940 * r178940;
        double r178942 = r178939 * r178941;
        double r178943 = r178938 - r178942;
        double r178944 = sqrt(r178943);
        double r178945 = r178937 * r178944;
        double r178946 = r178938 - r178941;
        double r178947 = r178945 * r178946;
        return r178947;
}
↓
double f(double v) {
        double r178948 = 2.0;
        double r178949 = sqrt(r178948);
        double r178950 = 4.0;
        double r178951 = r178949 / r178950;
        double r178952 = v;
        double r178953 = -r178952;
        double r178954 = 1.0;
        double r178955 = fma(r178953, r178952, r178954);
        double r178956 = 3.0;
        double r178957 = r178952 * r178952;
        double r178958 = r178956 * r178957;
        double r178959 = r178954 - r178958;
        double r178960 = sqrt(r178959);
        double r178961 = r178955 * r178960;
        double r178962 = r178951 * r178961;
        return r178962;
}

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. Using strategy rm

  3. Applied associate-*l*0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))