Average Error: 0.0 → 0.0
Time: 3.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot \frac{\sqrt{2}}{\frac{4}{1 - v \cdot v}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{\mathsf{fma}\left(\left(v \cdot v\right), -3, 1\right)} \cdot \frac{\sqrt{2}}{\frac{4}{1 - v \cdot v}}
double f(double v) {
        double r36188041 = 2.0;
        double r36188042 = sqrt(r36188041);
        double r36188043 = 4.0;
        double r36188044 = r36188042 / r36188043;
        double r36188045 = 1.0;
        double r36188046 = 3.0;
        double r36188047 = v;
        double r36188048 = r36188047 * r36188047;
        double r36188049 = r36188046 * r36188048;
        double r36188050 = r36188045 - r36188049;
        double r36188051 = sqrt(r36188050);
        double r36188052 = r36188044 * r36188051;
        double r36188053 = r36188045 - r36188048;
        double r36188054 = r36188052 * r36188053;
        return r36188054;
}
↓
double f(double v) {
        double r36188055 = v;
        double r36188056 = r36188055 * r36188055;
        double r36188057 = -3.0;
        double r36188058 = 1.0;
        double r36188059 = fma(r36188056, r36188057, r36188058);
        double r36188060 = sqrt(r36188059);
        double r36188061 = 2.0;
        double r36188062 = sqrt(r36188061);
        double r36188063 = 4.0;
        double r36188064 = r36188058 - r36188056;
        double r36188065 = r36188063 / r36188064;
        double r36188066 = r36188062 / r36188065;
        double r36188067 = r36188060 * r36188066;
        return r36188067;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))