Average Error: 0.5 → 0.5
Time: 4.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{{1}^{3} - {\left(5 \cdot \left(v \cdot v\right)\right)}^{3}}{\left(\left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right) + 1\right) + 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)
double f(double v) {
        double r292174 = 1.0;
        double r292175 = 5.0;
        double r292176 = v;
        double r292177 = r292176 * r292176;
        double r292178 = r292175 * r292177;
        double r292179 = r292174 - r292178;
        double r292180 = r292177 - r292174;
        double r292181 = r292179 / r292180;
        double r292182 = acos(r292181);
        return r292182;
}
↓
double f(double v) {
        double r292183 = 1.0;
        double r292184 = 3.0;
        double r292185 = pow(r292183, r292184);
        double r292186 = 5.0;
        double r292187 = v;
        double r292188 = r292187 * r292187;
        double r292189 = r292186 * r292188;
        double r292190 = pow(r292189, r292184);
        double r292191 = r292185 - r292190;
        double r292192 = r292189 + r292183;
        double r292193 = r292189 * r292192;
        double r292194 = r292183 * r292183;
        double r292195 = r292193 + r292194;
        double r292196 = r292188 - r292183;
        double r292197 = r292195 * r292196;
        double r292198 = r292191 / r292197;
        double r292199 = acos(r292198);
        return r292199;
}

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 flip3--0.5

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

  4. Applied associate-/l/0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020065 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))