Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}}} \cdot \sqrt{\frac{1 - x}{\sqrt{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{1}{\sqrt{1 + x}}} \cdot \sqrt{\frac{1 - x}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r15480 = 2.0;
        double r15481 = 1.0;
        double r15482 = x;
        double r15483 = r15481 - r15482;
        double r15484 = r15481 + r15482;
        double r15485 = r15483 / r15484;
        double r15486 = sqrt(r15485);
        double r15487 = atan(r15486);
        double r15488 = r15480 * r15487;
        return r15488;
}
↓
double f(double x) {
        double r15489 = 2.0;
        double r15490 = 1.0;
        double r15491 = 1.0;
        double r15492 = x;
        double r15493 = r15491 + r15492;
        double r15494 = sqrt(r15493);
        double r15495 = r15490 / r15494;
        double r15496 = sqrt(r15495);
        double r15497 = r15491 - r15492;
        double r15498 = r15497 / r15494;
        double r15499 = sqrt(r15498);
        double r15500 = r15496 * r15499;
        double r15501 = atan(r15500);
        double r15502 = r15489 * r15501;
        return r15502;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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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-sqr-sqrt0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Applied sqrt-prod0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))