Average Error: 0.0 → 0.0
Time: 13.1s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)} \cdot \left(\left(x \cdot \left(x - 1\right) + 1 \cdot 1\right) \cdot \left(x \cdot \left(x - 1\right) + 1 \cdot 1\right)\right)}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)} \cdot \left(\left(x \cdot \left(x - 1\right) + 1 \cdot 1\right) \cdot \left(x \cdot \left(x - 1\right) + 1 \cdot 1\right)\right)}\right)
double f(double x) {
        double r20326 = 2.0;
        double r20327 = 1.0;
        double r20328 = x;
        double r20329 = r20327 - r20328;
        double r20330 = r20327 + r20328;
        double r20331 = r20329 / r20330;
        double r20332 = sqrt(r20331);
        double r20333 = atan(r20332);
        double r20334 = r20326 * r20333;
        return r20334;
}
↓
double f(double x) {
        double r20335 = 2.0;
        double r20336 = 1.0;
        double r20337 = r20336 * r20336;
        double r20338 = x;
        double r20339 = r20338 * r20338;
        double r20340 = r20337 - r20339;
        double r20341 = 3.0;
        double r20342 = pow(r20336, r20341);
        double r20343 = pow(r20338, r20341);
        double r20344 = r20342 + r20343;
        double r20345 = r20344 * r20344;
        double r20346 = r20340 / r20345;
        double r20347 = r20338 - r20336;
        double r20348 = r20338 * r20347;
        double r20349 = r20348 + r20337;
        double r20350 = r20349 * r20349;
        double r20351 = r20346 * r20350;
        double r20352 = sqrt(r20351);
        double r20353 = atan(r20352);
        double r20354 = r20335 * r20353;
        return r20354;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}}}}\right)\]

  4. Applied add-cbrt-cube0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt[3]{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right) \cdot \left(1 - x\right)}}}{\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}}}\right)\]

  5. Applied cbrt-undiv0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{\frac{\left(\left(1 - x\right) \cdot \left(1 - x\right)\right) \cdot \left(1 - x\right)}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}}}}\right)\]

  6. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{1 - x}{1 + x}\right)}^{3}}}}\right)\]
  7. Using strategy rm

  8. Applied flip--0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}}}{1 + x}\right)}^{3}}}\right)\]

  9. Applied associate-/l/0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\frac{1 \cdot 1 - x \cdot x}{\left(1 + x\right) \cdot \left(1 + x\right)}\right)}}^{3}}}\right)\]
  10. Using strategy rm

  11. Applied flip3-+0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 \cdot 1 - x \cdot x}{\left(1 + x\right) \cdot \color{blue}{\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}}}\right)}^{3}}}\right)\]

  12. Applied flip3-+0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}} \cdot \frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)}}\right)}^{3}}}\right)\]

  13. Applied frac-times0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\left(\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\frac{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)}{\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)}}}\right)}^{3}}}\right)\]

  14. Applied associate-/r/0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{{\color{blue}{\left(\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)} \cdot \left(\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)\right)}}^{3}}}\right)\]

  15. Applied unpow-prod-down0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\sqrt[3]{\color{blue}{{\left(\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)}\right)}^{3} \cdot {\left(\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)}^{3}}}}\right)\]

  16. Applied cbrt-prod0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{{\left(\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)}\right)}^{3}} \cdot \sqrt[3]{{\left(\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)}^{3}}}}\right)\]

  17. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)}} \cdot \sqrt[3]{{\left(\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right)\right)}^{3}}}\right)\]

  18. Simplified0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)} \cdot \color{blue}{\left(\left(x \cdot \left(x - 1\right) + 1 \cdot 1\right) \cdot \left(x \cdot \left(x - 1\right) + 1 \cdot 1\right)\right)}}\right)\]

  19. Final simplification0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 \cdot 1 - x \cdot x}{\left({1}^{3} + {x}^{3}\right) \cdot \left({1}^{3} + {x}^{3}\right)} \cdot \left(\left(x \cdot \left(x - 1\right) + 1 \cdot 1\right) \cdot \left(x \cdot \left(x - 1\right) + 1 \cdot 1\right)\right)}\right)\]

Reproduce

herbie shell --seed 2019298 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))