Average Error: 0.0 → 0.0
Time: 26.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(-v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right) + \frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(-v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right) + \frac{\sqrt{2}}{4} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}
double f(double v) {
        double r3903564 = 2.0;
        double r3903565 = sqrt(r3903564);
        double r3903566 = 4.0;
        double r3903567 = r3903565 / r3903566;
        double r3903568 = 1.0;
        double r3903569 = 3.0;
        double r3903570 = v;
        double r3903571 = r3903570 * r3903570;
        double r3903572 = r3903569 * r3903571;
        double r3903573 = r3903568 - r3903572;
        double r3903574 = sqrt(r3903573);
        double r3903575 = r3903567 * r3903574;
        double r3903576 = r3903568 - r3903571;
        double r3903577 = r3903575 * r3903576;
        return r3903577;
}
↓
double f(double v) {
        double r3903578 = v;
        double r3903579 = r3903578 * r3903578;
        double r3903580 = -r3903579;
        double r3903581 = 2.0;
        double r3903582 = sqrt(r3903581);
        double r3903583 = 4.0;
        double r3903584 = r3903582 / r3903583;
        double r3903585 = 1.0;
        double r3903586 = 3.0;
        double r3903587 = r3903579 * r3903586;
        double r3903588 = r3903585 - r3903587;
        double r3903589 = sqrt(r3903588);
        double r3903590 = r3903584 * r3903589;
        double r3903591 = r3903580 * r3903590;
        double r3903592 = r3903591 + r3903590;
        return r3903592;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))