Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]
\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} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r278263 = 2.0;
        double r278264 = sqrt(r278263);
        double r278265 = 4.0;
        double r278266 = r278264 / r278265;
        double r278267 = 1.0;
        double r278268 = 3.0;
        double r278269 = v;
        double r278270 = r278269 * r278269;
        double r278271 = r278268 * r278270;
        double r278272 = r278267 - r278271;
        double r278273 = sqrt(r278272);
        double r278274 = r278266 * r278273;
        double r278275 = r278267 - r278270;
        double r278276 = r278274 * r278275;
        return r278276;
}
↓
double f(double v) {
        double r278277 = 2.0;
        double r278278 = sqrt(r278277);
        double r278279 = 1.0;
        double r278280 = r278279 * r278279;
        double r278281 = 3.0;
        double r278282 = v;
        double r278283 = r278282 * r278282;
        double r278284 = r278281 * r278283;
        double r278285 = r278284 * r278284;
        double r278286 = r278280 - r278285;
        double r278287 = sqrt(r278286);
        double r278288 = r278278 * r278287;
        double r278289 = 4.0;
        double r278290 = r278279 + r278284;
        double r278291 = sqrt(r278290);
        double r278292 = r278289 * r278291;
        double r278293 = r278288 / r278292;
        double r278294 = r278279 - r278283;
        double r278295 = r278293 * r278294;
        return r278295;
}

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 flip--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied frac-times0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019354 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))