Average Error: 0.5 → 1.0
Time: 4.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)\right)}^{3}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)\right)}^{3}}
double f(double v) {
        double r394487 = 1.0;
        double r394488 = 5.0;
        double r394489 = v;
        double r394490 = r394489 * r394489;
        double r394491 = r394488 * r394490;
        double r394492 = r394487 - r394491;
        double r394493 = r394490 - r394487;
        double r394494 = r394492 / r394493;
        double r394495 = acos(r394494);
        return r394495;
}
↓
double f(double v) {
        double r394496 = atan2(1.0, 0.0);
        double r394497 = 2.0;
        double r394498 = r394496 / r394497;
        double r394499 = 1.0;
        double r394500 = 5.0;
        double r394501 = v;
        double r394502 = r394501 * r394501;
        double r394503 = r394500 * r394502;
        double r394504 = r394499 - r394503;
        double r394505 = cbrt(r394504);
        double r394506 = r394505 * r394505;
        double r394507 = sqrt(r394499);
        double r394508 = r394501 + r394507;
        double r394509 = r394506 / r394508;
        double r394510 = r394501 - r394507;
        double r394511 = r394505 / r394510;
        double r394512 = r394509 * r394511;
        double r394513 = asin(r394512);
        double r394514 = 3.0;
        double r394515 = pow(r394513, r394514);
        double r394516 = cbrt(r394515);
        double r394517 = r394498 - r394516;
        return r394517;
}

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 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 add-sqr-sqrt0.5

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

  6. Applied difference-of-squares0.9

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

  7. Applied add-cube-cbrt1.0

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\frac{\color{blue}{\left(\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}}{\left(v + \sqrt{1}\right) \cdot \left(v - \sqrt{1}\right)}\right)\]

  8. Applied times-frac1.0

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube1.0

    \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt[3]{\left(\sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right) \cdot \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)\right) \cdot \sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)}}\]

  11. Simplified1.0

    \[\leadsto \frac{\pi}{2} - \sqrt[3]{\color{blue}{{\left(\sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)\right)}^{3}}}\]

  12. Final simplification1.0

    \[\leadsto \frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v + \sqrt{1}} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v - \sqrt{1}}\right)\right)}^{3}}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))