Average Error: 0.0 → 0.0
Time: 7.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{\frac{{v}^{2} \cdot \left({v}^{2} + 1\right) + 1 \cdot 1}{\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}} \cdot \frac{\sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{4}\]
\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}}{\frac{{v}^{2} \cdot \left({v}^{2} + 1\right) + 1 \cdot 1}{\left|\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}\right|}} \cdot \frac{\sqrt{\sqrt[3]{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{4}
double code(double v) {
	return ((double) (((double) (((double) (((double) sqrt(2.0)) / 4.0)) * ((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))) * ((double) (1.0 - ((double) (v * v))))));
}
↓
double code(double v) {
	return ((double) (((double) (((double) sqrt(2.0)) / ((double) (((double) (((double) (((double) pow(v, 2.0)) * ((double) (((double) pow(v, 2.0)) + 1.0)))) + ((double) (1.0 * 1.0)))) / ((double) fabs(((double) cbrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))))))) * ((double) (((double) (((double) sqrt(((double) cbrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))) * ((double) (((double) pow(1.0, 3.0)) - ((double) pow(((double) (v * v)), 3.0)))))) / 4.0))));
}

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 flip3--0.0

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

  4. Applied associate-*l/0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  9. Applied add-cube-cbrt0.0

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

  10. Applied sqrt-prod0.0

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

  11. Applied associate-*l*0.0

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

  12. Applied associate-*r*0.0

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

  13. Applied times-frac0.0

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

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020114 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))