Average Error: 0.5 → 0.6
Time: 28.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}
double f(double v) {
        double r143978 = 1.0;
        double r143979 = 5.0;
        double r143980 = v;
        double r143981 = r143980 * r143980;
        double r143982 = r143979 * r143981;
        double r143983 = r143978 - r143982;
        double r143984 = r143981 - r143978;
        double r143985 = r143983 / r143984;
        double r143986 = acos(r143985);
        return r143986;
}
↓
double f(double v) {
        double r143987 = 1.0;
        double r143988 = 5.0;
        double r143989 = v;
        double r143990 = r143989 * r143989;
        double r143991 = r143988 * r143990;
        double r143992 = exp(r143991);
        double r143993 = log(r143992);
        double r143994 = r143987 - r143993;
        double r143995 = r143990 - r143987;
        double r143996 = r143994 / r143995;
        double r143997 = acos(r143996);
        double r143998 = log(r143997);
        double r143999 = sqrt(r143998);
        double r144000 = r143987 - r143991;
        double r144001 = r144000 / r143995;
        double r144002 = acos(r144001);
        double r144003 = log(r144002);
        double r144004 = sqrt(r144003);
        double r144005 = r143999 * r144004;
        double r144006 = exp(r144005);
        return r144006;
}

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 add-exp-log0.5

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

  5. Applied add-sqr-sqrt0.6

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

  7. Applied add-log-exp0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))