Average Error: 0.6 → 0.6
Time: 26.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\right)}
double f(double v) {
        double r117203 = 1.0;
        double r117204 = 5.0;
        double r117205 = v;
        double r117206 = r117205 * r117205;
        double r117207 = r117204 * r117206;
        double r117208 = r117203 - r117207;
        double r117209 = r117206 - r117203;
        double r117210 = r117208 / r117209;
        double r117211 = acos(r117210);
        return r117211;
}
↓
double f(double v) {
        double r117212 = 1.0;
        double r117213 = 5.0;
        double r117214 = v;
        double r117215 = 2.0;
        double r117216 = pow(r117214, r117215);
        double r117217 = r117213 * r117216;
        double r117218 = r117212 - r117217;
        double r117219 = 6.0;
        double r117220 = pow(r117214, r117219);
        double r117221 = 3.0;
        double r117222 = pow(r117212, r117221);
        double r117223 = r117220 - r117222;
        double r117224 = r117218 / r117223;
        double r117225 = r117214 * r117214;
        double r117226 = r117212 + r117225;
        double r117227 = r117212 * r117226;
        double r117228 = 4.0;
        double r117229 = pow(r117214, r117228);
        double r117230 = r117227 + r117229;
        double r117231 = r117224 * r117230;
        double r117232 = acos(r117231);
        double r117233 = log(r117232);
        double r117234 = sqrt(r117233);
        double r117235 = exp(r117234);
        double r117236 = pow(r117235, r117234);
        return r117236;
}

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 flip3--0.6

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

  4. Applied associate-/r/0.6

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

  5. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{6} - {1}^{3}}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]
  6. Using strategy rm

  7. Applied add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{6} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)}}\]

  8. Simplified0.6

    \[\leadsto e^{\color{blue}{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.6

    \[\leadsto e^{\color{blue}{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}}\]

  11. Applied exp-prod0.6

    \[\leadsto \color{blue}{{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\right)}}\]

  12. Final simplification0.6

    \[\leadsto {\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\right)}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))