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 r288208 = 1.0;
        double r288209 = 5.0;
        double r288210 = v;
        double r288211 = r288210 * r288210;
        double r288212 = r288209 * r288211;
        double r288213 = r288208 - r288212;
        double r288214 = r288211 - r288208;
        double r288215 = r288213 / r288214;
        double r288216 = acos(r288215);
        return r288216;
}
↓
double f(double v) {
        double r288217 = 5.0;
        double r288218 = r288217 * r288217;
        double r288219 = -r288218;
        double r288220 = v;
        double r288221 = 4.0;
        double r288222 = pow(r288220, r288221);
        double r288223 = r288219 * r288222;
        double r288224 = 1.0;
        double r288225 = r288224 * r288224;
        double r288226 = r288223 + r288225;
        double r288227 = r288220 * r288220;
        double r288228 = r288217 * r288227;
        double r288229 = r288224 + r288228;
        double r288230 = r288226 / r288229;
        double r288231 = r288227 - r288224;
        double r288232 = r288230 / r288231;
        double r288233 = acos(r288232);
        return r288233;
}

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 2019354 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))