Average Error: 0.6 → 0.6
Time: 14.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r308924 = 1.0;
        double r308925 = 5.0;
        double r308926 = v;
        double r308927 = r308926 * r308926;
        double r308928 = r308925 * r308927;
        double r308929 = r308924 - r308928;
        double r308930 = r308927 - r308924;
        double r308931 = r308929 / r308930;
        double r308932 = acos(r308931);
        return r308932;
}
↓
double f(double v) {
        double r308933 = 1.0;
        double r308934 = r308933 * r308933;
        double r308935 = 5.0;
        double r308936 = r308935 * r308935;
        double r308937 = v;
        double r308938 = 4.0;
        double r308939 = pow(r308937, r308938);
        double r308940 = r308936 * r308939;
        double r308941 = r308934 - r308940;
        double r308942 = r308937 * r308937;
        double r308943 = r308935 * r308942;
        double r308944 = r308933 + r308943;
        double r308945 = r308941 / r308944;
        double r308946 = r308942 - r308933;
        double r308947 = r308945 / r308946;
        double r308948 = acos(r308947);
        return r308948;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied flip--0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{1 + 5 \cdot \left(v \cdot v\right)}}}{v \cdot v - 1}\right)\]

  4. Simplified0.6

    \[\leadsto \cos^{-1} \left(\frac{\frac{\color{blue}{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

  5. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\frac{\frac{1 \cdot 1 - \left(5 \cdot 5\right) \cdot {v}^{4}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2020042 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))