Average Error: 0.0 → 0.0
Time: 5.3s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2} \cdot \left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}{4} \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\frac{\sqrt{2} \cdot \left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}{4} \cdot \sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r275232 = 2.0;
        double r275233 = sqrt(r275232);
        double r275234 = 4.0;
        double r275235 = r275233 / r275234;
        double r275236 = 1.0;
        double r275237 = 3.0;
        double r275238 = v;
        double r275239 = r275238 * r275238;
        double r275240 = r275237 * r275239;
        double r275241 = r275236 - r275240;
        double r275242 = sqrt(r275241);
        double r275243 = r275235 * r275242;
        double r275244 = r275236 - r275239;
        double r275245 = r275243 * r275244;
        return r275245;
}
↓
double f(double v) {
        double r275246 = 2.0;
        double r275247 = sqrt(r275246);
        double r275248 = 1.0;
        double r275249 = 3.0;
        double r275250 = v;
        double r275251 = r275250 * r275250;
        double r275252 = r275249 * r275251;
        double r275253 = r275248 - r275252;
        double r275254 = cbrt(r275253);
        double r275255 = fabs(r275254);
        double r275256 = r275247 * r275255;
        double r275257 = 4.0;
        double r275258 = r275256 / r275257;
        double r275259 = sqrt(r275254);
        double r275260 = r275258 * r275259;
        double r275261 = r275248 - r275251;
        double r275262 = r275260 * r275261;
        return r275262;
}

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-cube-cbrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Applied associate-*r*0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019318 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))