Average Error: 0.6 → 0.6
Time: 4.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\log \left(e^{1 - 5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\log \left(e^{1 - 5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r317138 = 1.0;
        double r317139 = 5.0;
        double r317140 = v;
        double r317141 = r317140 * r317140;
        double r317142 = r317139 * r317141;
        double r317143 = r317138 - r317142;
        double r317144 = r317141 - r317138;
        double r317145 = r317143 / r317144;
        double r317146 = acos(r317145);
        return r317146;
}
↓
double f(double v) {
        double r317147 = 1.0;
        double r317148 = 5.0;
        double r317149 = v;
        double r317150 = r317149 * r317149;
        double r317151 = r317148 * r317150;
        double r317152 = r317147 - r317151;
        double r317153 = exp(r317152);
        double r317154 = log(r317153);
        double r317155 = r317150 - r317147;
        double r317156 = r317154 / r317155;
        double r317157 = acos(r317156);
        return r317157;
}

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 add-log-exp0.6

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

  4. Applied add-log-exp0.6

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

  5. Applied diff-log0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020027 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))