Average Error: 0.0 → 0.0
Time: 22.4s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)
double f(double v) {
        double r2567585 = 2.0;
        double r2567586 = sqrt(r2567585);
        double r2567587 = 4.0;
        double r2567588 = r2567586 / r2567587;
        double r2567589 = 1.0;
        double r2567590 = 3.0;
        double r2567591 = v;
        double r2567592 = r2567591 * r2567591;
        double r2567593 = r2567590 * r2567592;
        double r2567594 = r2567589 - r2567593;
        double r2567595 = sqrt(r2567594);
        double r2567596 = r2567588 * r2567595;
        double r2567597 = r2567589 - r2567592;
        double r2567598 = r2567596 * r2567597;
        return r2567598;
}
↓
double f(double v) {
        double r2567599 = 1.0;
        double r2567600 = v;
        double r2567601 = r2567600 * r2567600;
        double r2567602 = r2567599 - r2567601;
        double r2567603 = 3.0;
        double r2567604 = r2567601 * r2567603;
        double r2567605 = r2567599 - r2567604;
        double r2567606 = sqrt(r2567605);
        double r2567607 = sqrt(r2567606);
        double r2567608 = r2567607 * r2567607;
        double r2567609 = 2.0;
        double r2567610 = sqrt(r2567609);
        double r2567611 = 4.0;
        double r2567612 = r2567610 / r2567611;
        double r2567613 = r2567608 * r2567612;
        double r2567614 = r2567602 * r2567613;
        return r2567614;
}

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 add-sqr-sqrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))