Average Error: 0.6 → 0.7
Time: 15.5s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)
double f(double v) {
        double r303537 = 1.0;
        double r303538 = 5.0;
        double r303539 = v;
        double r303540 = r303539 * r303539;
        double r303541 = r303538 * r303540;
        double r303542 = r303537 - r303541;
        double r303543 = r303540 - r303537;
        double r303544 = r303542 / r303543;
        double r303545 = acos(r303544);
        return r303545;
}
↓
double f(double v) {
        double r303546 = atan2(1.0, 0.0);
        double r303547 = 2.0;
        double r303548 = r303546 / r303547;
        double r303549 = 4.0;
        double r303550 = v;
        double r303551 = r303550 * r303550;
        double r303552 = 4.0;
        double r303553 = pow(r303550, r303552);
        double r303554 = r303551 + r303553;
        double r303555 = r303549 * r303554;
        double r303556 = 1.0;
        double r303557 = r303555 - r303556;
        double r303558 = asin(r303557);
        double r303559 = r303548 - r303558;
        return r303559;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\]

  3. Simplified0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)}\]
  4. Using strategy rm

  5. Applied acos-asin0.7

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)}\]

  6. Final simplification0.7

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(4 \cdot \left(v \cdot v + {v}^{4}\right) - 1\right)\]

Reproduce

herbie shell --seed 2020045 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))