Average Error: 0.0 → 0.0
Time: 3.5m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(1 - v \cdot v\right)\]
double f(double v) {
        double r49604421 = 2.0;
        double r49604422 = sqrt(r49604421);
        double r49604423 = 4.0;
        double r49604424 = r49604422 / r49604423;
        double r49604425 = 1.0;
        double r49604426 = 3.0;
        double r49604427 = v;
        double r49604428 = r49604427 * r49604427;
        double r49604429 = r49604426 * r49604428;
        double r49604430 = r49604425 - r49604429;
        double r49604431 = sqrt(r49604430);
        double r49604432 = r49604424 * r49604431;
        double r49604433 = r49604425 - r49604428;
        double r49604434 = r49604432 * r49604433;
        return r49604434;
}
↓
double f(double v) {
        double r49604435 = 1.0;
        double r49604436 = v;
        double r49604437 = r49604436 * r49604436;
        double r49604438 = 3.0;
        double r49604439 = r49604437 * r49604438;
        double r49604440 = r49604435 - r49604439;
        double r49604441 = sqrt(r49604440);
        double r49604442 = 2.0;
        double r49604443 = sqrt(r49604442);
        double r49604444 = 4.0;
        double r49604445 = r49604443 / r49604444;
        double r49604446 = r49604441 * r49604445;
        double r49604447 = r49604435 - r49604437;
        double r49604448 = r49604446 * r49604447;
        return r49604448;
}


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

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))