Average Error: 0.0 → 0.0
Time: 16.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 \cdot 1 - {v}^{4}}}{\sqrt{1 + v \cdot v}} \cdot \sqrt{1 - v \cdot v}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 \cdot 1 - {v}^{4}}}{\sqrt{1 + v \cdot v}} \cdot \sqrt{1 - v \cdot v}
double f(double v) {
        double r229248 = 2.0;
        double r229249 = sqrt(r229248);
        double r229250 = 4.0;
        double r229251 = r229249 / r229250;
        double r229252 = 1.0;
        double r229253 = 3.0;
        double r229254 = v;
        double r229255 = r229254 * r229254;
        double r229256 = r229253 * r229255;
        double r229257 = r229252 - r229256;
        double r229258 = sqrt(r229257);
        double r229259 = r229251 * r229258;
        double r229260 = r229252 - r229255;
        double r229261 = r229259 * r229260;
        return r229261;
}
↓
double f(double v) {
        double r229262 = 2.0;
        double r229263 = sqrt(r229262);
        double r229264 = 4.0;
        double r229265 = r229263 / r229264;
        double r229266 = 1.0;
        double r229267 = 3.0;
        double r229268 = v;
        double r229269 = r229268 * r229268;
        double r229270 = r229267 * r229269;
        double r229271 = r229266 - r229270;
        double r229272 = sqrt(r229271);
        double r229273 = r229265 * r229272;
        double r229274 = r229266 * r229266;
        double r229275 = 4.0;
        double r229276 = pow(r229268, r229275);
        double r229277 = r229274 - r229276;
        double r229278 = sqrt(r229277);
        double r229279 = r229273 * r229278;
        double r229280 = r229266 + r229269;
        double r229281 = sqrt(r229280);
        double r229282 = r229279 / r229281;
        double r229283 = r229266 - r229269;
        double r229284 = sqrt(r229283);
        double r229285 = r229282 * r229284;
        return r229285;
}

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

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

  4. Applied associate-*r*0.0

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

  6. Applied flip--0.0

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

  7. Applied sqrt-div0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))