Average Error: 0.5 → 1.0
Time: 4.5s
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 r303419 = 1.0;
        double r303420 = 5.0;
        double r303421 = v;
        double r303422 = r303421 * r303421;
        double r303423 = r303420 * r303422;
        double r303424 = r303419 - r303423;
        double r303425 = r303422 - r303419;
        double r303426 = r303424 / r303425;
        double r303427 = acos(r303426);
        return r303427;
}
↓
double f(double v) {
        double r303428 = atan2(1.0, 0.0);
        double r303429 = 2.0;
        double r303430 = r303428 / r303429;
        double r303431 = 1.0;
        double r303432 = 5.0;
        double r303433 = v;
        double r303434 = r303433 * r303433;
        double r303435 = r303432 * r303434;
        double r303436 = r303431 - r303435;
        double r303437 = cbrt(r303436);
        double r303438 = r303437 * r303437;
        double r303439 = sqrt(r303431);
        double r303440 = r303433 + r303439;
        double r303441 = r303438 / r303440;
        double r303442 = r303433 - r303439;
        double r303443 = r303437 / r303442;
        double r303444 = r303441 * r303443;
        double r303445 = asin(r303444);
        double r303446 = 3.0;
        double r303447 = pow(r303445, r303446);
        double r303448 = cbrt(r303447);
        double r303449 = r303430 - r303448;
        return r303449;
}

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))))