Average Error: 0.5 → 0.5
Time: 4.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \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)\right)\right)
double f(double v) {
        double r304005 = 1.0;
        double r304006 = 5.0;
        double r304007 = v;
        double r304008 = r304007 * r304007;
        double r304009 = r304006 * r304008;
        double r304010 = r304005 - r304009;
        double r304011 = r304008 - r304005;
        double r304012 = r304010 / r304011;
        double r304013 = acos(r304012);
        return r304013;
}
↓
double f(double v) {
        double r304014 = 1.0;
        double r304015 = 5.0;
        double r304016 = v;
        double r304017 = r304016 * r304016;
        double r304018 = r304015 * r304017;
        double r304019 = r304014 - r304018;
        double r304020 = 3.0;
        double r304021 = pow(r304017, r304020);
        double r304022 = pow(r304014, r304020);
        double r304023 = r304021 - r304022;
        double r304024 = r304019 / r304023;
        double r304025 = r304017 * r304017;
        double r304026 = r304014 * r304014;
        double r304027 = r304017 * r304014;
        double r304028 = r304026 + r304027;
        double r304029 = r304025 + r304028;
        double r304030 = r304024 * r304029;
        double r304031 = acos(r304030);
        double r304032 = log1p(r304031);
        double r304033 = expm1(r304032);
        return r304033;
}

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 expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)}\]

  7. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))