Average Error: 0.5 → 0.5
Time: 5.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{3} \cdot {v}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]
\cos^{-1} \left(\frac{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}^{3} \cdot {v}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)
double f(double v) {
        double r315023 = 1.0;
        double r315024 = 5.0;
        double r315025 = v;
        double r315026 = r315025 * r315025;
        double r315027 = r315024 * r315026;
        double r315028 = r315023 - r315027;
        double r315029 = r315026 - r315023;
        double r315030 = r315028 / r315029;
        double r315031 = acos(r315030);
        return r315031;
}
↓
double f(double v) {
        double r315032 = 1.0;
        double r315033 = 5.0;
        double r315034 = v;
        double r315035 = r315034 * r315034;
        double r315036 = r315033 * r315035;
        double r315037 = r315032 - r315036;
        double r315038 = 3.0;
        double r315039 = pow(r315034, r315038);
        double r315040 = r315039 * r315039;
        double r315041 = pow(r315032, r315038);
        double r315042 = r315040 - r315041;
        double r315043 = r315037 / r315042;
        double r315044 = r315035 * r315035;
        double r315045 = r315032 * r315032;
        double r315046 = r315035 * r315032;
        double r315047 = r315045 + r315046;
        double r315048 = r315044 + r315047;
        double r315049 = r315043 * r315048;
        double r315050 = acos(r315049);
        return r315050;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  3. Applied flip3--0.5

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{{\left(v \cdot v\right)}^{3} - {1}^{3}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)}}}\right)\]

  4. Applied associate-/r/0.5

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)}\]
  5. Using strategy rm

  6. Applied unpow-prod-down0.5

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{{v}^{3} \cdot {v}^{3}} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]

  7. Final simplification0.5

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{3} \cdot {v}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]

Reproduce

herbie shell --seed 2020081 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))