Average Error: 0.0 → 0.0
Time: 15.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - v \cdot v}\right) \cdot \sqrt{1 - v \cdot v}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2}}{4} \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - v \cdot v}\right) \cdot \sqrt{1 - v \cdot v}\right)
double f(double v) {
        double r299560 = 2.0;
        double r299561 = sqrt(r299560);
        double r299562 = 4.0;
        double r299563 = r299561 / r299562;
        double r299564 = 1.0;
        double r299565 = 3.0;
        double r299566 = v;
        double r299567 = r299566 * r299566;
        double r299568 = r299565 * r299567;
        double r299569 = r299564 - r299568;
        double r299570 = sqrt(r299569);
        double r299571 = r299563 * r299570;
        double r299572 = r299564 - r299567;
        double r299573 = r299571 * r299572;
        return r299573;
}
↓
double f(double v) {
        double r299574 = 2.0;
        double r299575 = sqrt(r299574);
        double r299576 = 4.0;
        double r299577 = r299575 / r299576;
        double r299578 = 1.0;
        double r299579 = 3.0;
        double r299580 = v;
        double r299581 = r299580 * r299580;
        double r299582 = r299579 * r299581;
        double r299583 = r299578 - r299582;
        double r299584 = sqrt(r299583);
        double r299585 = r299578 - r299581;
        double r299586 = sqrt(r299585);
        double r299587 = r299584 * r299586;
        double r299588 = r299587 * r299586;
        double r299589 = r299577 * r299588;
        return r299589;
}

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 associate-*l*0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied associate-*r*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +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))))