Average Error: 0.6 → 0.6
Time: 5.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1}}{v \cdot v - 1}\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(\frac{\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1}}{v \cdot v - 1}\right)\right)}
double f(double v) {
        double r331521 = 1.0;
        double r331522 = 5.0;
        double r331523 = v;
        double r331524 = r331523 * r331523;
        double r331525 = r331522 * r331524;
        double r331526 = r331521 - r331525;
        double r331527 = r331524 - r331521;
        double r331528 = r331526 / r331527;
        double r331529 = acos(r331528);
        return r331529;
}
↓
double f(double v) {
        double r331530 = 1.0;
        double r331531 = 3.0;
        double r331532 = pow(r331530, r331531);
        double r331533 = 5.0;
        double r331534 = v;
        double r331535 = r331534 * r331534;
        double r331536 = r331533 * r331535;
        double r331537 = pow(r331536, r331531);
        double r331538 = r331532 - r331537;
        double r331539 = r331536 + r331530;
        double r331540 = r331536 * r331539;
        double r331541 = r331530 * r331530;
        double r331542 = r331540 + r331541;
        double r331543 = r331538 / r331542;
        double r331544 = r331535 - r331530;
        double r331545 = r331543 / r331544;
        double r331546 = acos(r331545);
        double r331547 = log(r331546);
        double r331548 = exp(r331547);
        return r331548;
}

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

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

  4. Simplified0.6

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

  6. Applied add-exp-log0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020100 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))