Average Error: 0.0 → 0.0
Time: 20.2s
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 r7278318 = 2.0;
        double r7278319 = sqrt(r7278318);
        double r7278320 = 4.0;
        double r7278321 = r7278319 / r7278320;
        double r7278322 = 1.0;
        double r7278323 = 3.0;
        double r7278324 = v;
        double r7278325 = r7278324 * r7278324;
        double r7278326 = r7278323 * r7278325;
        double r7278327 = r7278322 - r7278326;
        double r7278328 = sqrt(r7278327);
        double r7278329 = r7278321 * r7278328;
        double r7278330 = r7278322 - r7278325;
        double r7278331 = r7278329 * r7278330;
        return r7278331;
}
↓
double f(double v) {
        double r7278332 = 1.0;
        double r7278333 = v;
        double r7278334 = r7278333 * r7278333;
        double r7278335 = r7278332 - r7278334;
        double r7278336 = 3.0;
        double r7278337 = r7278334 * r7278336;
        double r7278338 = r7278332 - r7278337;
        double r7278339 = sqrt(r7278338);
        double r7278340 = sqrt(r7278339);
        double r7278341 = r7278340 * r7278340;
        double r7278342 = 2.0;
        double r7278343 = sqrt(r7278342);
        double r7278344 = 4.0;
        double r7278345 = r7278343 / r7278344;
        double r7278346 = r7278341 * r7278345;
        double r7278347 = r7278335 * r7278346;
        return r7278347;
}

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 2019170 +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))))