Average Error: 0.0 → 0.0
Time: 21.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[1 \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) + \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(-v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
1 \cdot \left(\frac{\sqrt{2}}{4} \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(-v \cdot v\right)
double f(double v) {
        double r201032 = 2.0;
        double r201033 = sqrt(r201032);
        double r201034 = 4.0;
        double r201035 = r201033 / r201034;
        double r201036 = 1.0;
        double r201037 = 3.0;
        double r201038 = v;
        double r201039 = r201038 * r201038;
        double r201040 = r201037 * r201039;
        double r201041 = r201036 - r201040;
        double r201042 = sqrt(r201041);
        double r201043 = r201035 * r201042;
        double r201044 = r201036 - r201039;
        double r201045 = r201043 * r201044;
        return r201045;
}
↓
double f(double v) {
        double r201046 = 1.0;
        double r201047 = 2.0;
        double r201048 = sqrt(r201047);
        double r201049 = 4.0;
        double r201050 = r201048 / r201049;
        double r201051 = 3.0;
        double r201052 = v;
        double r201053 = r201052 * r201052;
        double r201054 = r201051 * r201053;
        double r201055 = r201046 - r201054;
        double r201056 = sqrt(r201055);
        double r201057 = r201050 * r201056;
        double r201058 = r201046 * r201057;
        double r201059 = -r201053;
        double r201060 = r201057 * r201059;
        double r201061 = r201058 + r201060;
        return r201061;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 sub-neg0.0

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

  4. Applied distribute-rgt-in0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +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))))