Average Error: 0.5 → 1.0
Time: 5.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)
double f(double v) {
        double r221307 = 1.0;
        double r221308 = 5.0;
        double r221309 = v;
        double r221310 = r221309 * r221309;
        double r221311 = r221308 * r221310;
        double r221312 = r221307 - r221311;
        double r221313 = r221310 - r221307;
        double r221314 = r221312 / r221313;
        double r221315 = acos(r221314);
        return r221315;
}
↓
double f(double v) {
        double r221316 = 1.0;
        double r221317 = 5.0;
        double r221318 = v;
        double r221319 = r221318 * r221318;
        double r221320 = r221317 * r221319;
        double r221321 = r221316 - r221320;
        double r221322 = sqrt(r221321);
        double r221323 = sqrt(r221316);
        double r221324 = r221318 + r221323;
        double r221325 = r221322 / r221324;
        double r221326 = r221318 - r221323;
        double r221327 = r221322 / r221326;
        double r221328 = r221325 * r221327;
        double r221329 = acos(r221328);
        return r221329;
}

Error

Bits error versus v

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Your Program's Arguments

Results

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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 add-sqr-sqrt0.5

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

  4. Applied difference-of-squares1.0

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

  5. Applied add-sqr-sqrt1.0

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

  6. Applied times-frac1.0

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

  7. Final simplification1.0

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))