Average Error: 0.0 → 0.0
Time: 4.8s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\frac{\sqrt{2}}{4} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\frac{\sqrt{2}}{4} \cdot \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)}} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r146188 = 2.0;
        double r146189 = sqrt(r146188);
        double r146190 = 4.0;
        double r146191 = r146189 / r146190;
        double r146192 = 1.0;
        double r146193 = 3.0;
        double r146194 = v;
        double r146195 = r146194 * r146194;
        double r146196 = r146193 * r146195;
        double r146197 = r146192 - r146196;
        double r146198 = sqrt(r146197);
        double r146199 = r146191 * r146198;
        double r146200 = r146192 - r146195;
        double r146201 = r146199 * r146200;
        return r146201;
}
↓
double f(double v) {
        double r146202 = 2.0;
        double r146203 = sqrt(r146202);
        double r146204 = 4.0;
        double r146205 = r146203 / r146204;
        double r146206 = 1.0;
        double r146207 = r146206 * r146206;
        double r146208 = 3.0;
        double r146209 = v;
        double r146210 = r146209 * r146209;
        double r146211 = r146208 * r146210;
        double r146212 = r146211 * r146211;
        double r146213 = r146207 - r146212;
        double r146214 = sqrt(r146213);
        double r146215 = r146205 * r146214;
        double r146216 = r146206 + r146211;
        double r146217 = sqrt(r146216);
        double r146218 = r146215 / r146217;
        double r146219 = r146206 - r146210;
        double r146220 = r146218 * r146219;
        return r146220;
}

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 associate-*r/0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))