Average Error: 0.5 → 0.5
Time: 5.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\frac{\left(-5 \cdot 5\right) \cdot {v}^{4} + 1 \cdot 1}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\frac{\left(-5 \cdot 5\right) \cdot {v}^{4} + 1 \cdot 1}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r330194 = 1.0;
        double r330195 = 5.0;
        double r330196 = v;
        double r330197 = r330196 * r330196;
        double r330198 = r330195 * r330197;
        double r330199 = r330194 - r330198;
        double r330200 = r330197 - r330194;
        double r330201 = r330199 / r330200;
        double r330202 = acos(r330201);
        return r330202;
}
↓
double f(double v) {
        double r330203 = 5.0;
        double r330204 = r330203 * r330203;
        double r330205 = -r330204;
        double r330206 = v;
        double r330207 = 4.0;
        double r330208 = pow(r330206, r330207);
        double r330209 = r330205 * r330208;
        double r330210 = 1.0;
        double r330211 = r330210 * r330210;
        double r330212 = r330209 + r330211;
        double r330213 = r330206 * r330206;
        double r330214 = r330203 * r330213;
        double r330215 = r330210 + r330214;
        double r330216 = r330212 / r330215;
        double r330217 = r330213 - r330210;
        double r330218 = r330216 / r330217;
        double r330219 = acos(r330218);
        return r330219;
}

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 flip--0.5

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

  4. Simplified0.5

    \[\leadsto \cos^{-1} \left(\frac{\frac{\color{blue}{\left(-5 \cdot 5\right) \cdot {v}^{4} + 1 \cdot 1}}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

  5. Final simplification0.5

    \[\leadsto \cos^{-1} \left(\frac{\frac{\left(-5 \cdot 5\right) \cdot {v}^{4} + 1 \cdot 1}{1 + 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2019356 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))