Average Error: 0.0 → 0.0
Time: 12.2s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \frac{1 - x}{{1}^{3} + {x}^{3}}}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{\left(1 \cdot 1 + \left(x \cdot x - 1 \cdot x\right)\right) \cdot \frac{1 - x}{{1}^{3} + {x}^{3}}}\right) \cdot 2
double f(double x) {
        double r23121 = 2.0;
        double r23122 = 1.0;
        double r23123 = x;
        double r23124 = r23122 - r23123;
        double r23125 = r23122 + r23123;
        double r23126 = r23124 / r23125;
        double r23127 = sqrt(r23126);
        double r23128 = atan(r23127);
        double r23129 = r23121 * r23128;
        return r23129;
}
↓
double f(double x) {
        double r23130 = 1.0;
        double r23131 = r23130 * r23130;
        double r23132 = x;
        double r23133 = r23132 * r23132;
        double r23134 = r23130 * r23132;
        double r23135 = r23133 - r23134;
        double r23136 = r23131 + r23135;
        double r23137 = r23130 - r23132;
        double r23138 = 3.0;
        double r23139 = pow(r23130, r23138);
        double r23140 = pow(r23132, r23138);
        double r23141 = r23139 + r23140;
        double r23142 = r23137 / r23141;
        double r23143 = r23136 * r23142;
        double r23144 = sqrt(r23143);
        double r23145 = atan(r23144);
        double r23146 = 2.0;
        double r23147 = r23145 * r23146;
        return r23147;
}

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. Simplified0.0

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "arccos"
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))