Average Error: 0.5 → 0.5
Time: 22.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)
double f(double v) {
        double r170381 = 1.0;
        double r170382 = 5.0;
        double r170383 = v;
        double r170384 = r170383 * r170383;
        double r170385 = r170382 * r170384;
        double r170386 = r170381 - r170385;
        double r170387 = r170384 - r170381;
        double r170388 = r170386 / r170387;
        double r170389 = acos(r170388);
        return r170389;
}
↓
double f(double v) {
        double r170390 = atan2(1.0, 0.0);
        double r170391 = 2.0;
        double r170392 = r170390 / r170391;
        double r170393 = 1.0;
        double r170394 = 5.0;
        double r170395 = v;
        double r170396 = pow(r170395, r170391);
        double r170397 = r170394 * r170396;
        double r170398 = r170393 - r170397;
        double r170399 = r170396 - r170393;
        double r170400 = r170398 / r170399;
        double r170401 = asin(r170400);
        double r170402 = -r170401;
        double r170403 = r170392 + r170402;
        return r170403;
}

Error

Bits error versus v

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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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied sub-neg0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}\]

  6. Simplified0.5

    \[\leadsto \frac{\pi}{2} + \color{blue}{\left(-\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}\]

  7. Final simplification0.5

    \[\leadsto \frac{\pi}{2} + \left(-\sin^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))