Average Error: 0.0 → 0.0
Time: 13.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}^{3}} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}^{3}} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r155942 = 2.0;
        double r155943 = sqrt(r155942);
        double r155944 = 4.0;
        double r155945 = r155943 / r155944;
        double r155946 = 1.0;
        double r155947 = 3.0;
        double r155948 = v;
        double r155949 = r155948 * r155948;
        double r155950 = r155947 * r155949;
        double r155951 = r155946 - r155950;
        double r155952 = sqrt(r155951);
        double r155953 = r155945 * r155952;
        double r155954 = r155946 - r155949;
        double r155955 = r155953 * r155954;
        return r155955;
}
↓
double f(double v) {
        double r155956 = 2.0;
        double r155957 = sqrt(r155956);
        double r155958 = 4.0;
        double r155959 = r155957 / r155958;
        double r155960 = 1.0;
        double r155961 = 3.0;
        double r155962 = v;
        double r155963 = r155962 * r155962;
        double r155964 = r155961 * r155963;
        double r155965 = r155960 - r155964;
        double r155966 = sqrt(r155965);
        double r155967 = r155959 * r155966;
        double r155968 = 3.0;
        double r155969 = pow(r155967, r155968);
        double r155970 = cbrt(r155969);
        double r155971 = r155960 - r155963;
        double r155972 = r155970 * r155971;
        return r155972;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-undiv0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))