Average Error: 0.0 → 0.0
Time: 20.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\sqrt{1} - v\right) \cdot \left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\sqrt{1} + v\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\sqrt{1} - v\right) \cdot \left(\left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \frac{\sqrt{2}}{4}\right) \cdot \left(\sqrt{1} + v\right)\right)
double f(double v) {
        double r8068589 = 2.0;
        double r8068590 = sqrt(r8068589);
        double r8068591 = 4.0;
        double r8068592 = r8068590 / r8068591;
        double r8068593 = 1.0;
        double r8068594 = 3.0;
        double r8068595 = v;
        double r8068596 = r8068595 * r8068595;
        double r8068597 = r8068594 * r8068596;
        double r8068598 = r8068593 - r8068597;
        double r8068599 = sqrt(r8068598);
        double r8068600 = r8068592 * r8068599;
        double r8068601 = r8068593 - r8068596;
        double r8068602 = r8068600 * r8068601;
        return r8068602;
}
↓
double f(double v) {
        double r8068603 = 1.0;
        double r8068604 = sqrt(r8068603);
        double r8068605 = v;
        double r8068606 = r8068604 - r8068605;
        double r8068607 = r8068605 * r8068605;
        double r8068608 = 3.0;
        double r8068609 = r8068607 * r8068608;
        double r8068610 = r8068603 - r8068609;
        double r8068611 = sqrt(r8068610);
        double r8068612 = 2.0;
        double r8068613 = sqrt(r8068612);
        double r8068614 = 4.0;
        double r8068615 = r8068613 / r8068614;
        double r8068616 = r8068611 * r8068615;
        double r8068617 = r8068604 + r8068605;
        double r8068618 = r8068616 * r8068617;
        double r8068619 = r8068606 * r8068618;
        return r8068619;
}

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

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

  4. Applied difference-of-squares0.0

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

  5. Applied associate-*r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))