Average Error: 0.0 → 0.0
Time: 3.2s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt{1 + x}}}\right)\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt{1 + x}}}\right)
double f(double x) {
        double r8574 = 2.0;
        double r8575 = 1.0;
        double r8576 = x;
        double r8577 = r8575 - r8576;
        double r8578 = r8575 + r8576;
        double r8579 = r8577 / r8578;
        double r8580 = sqrt(r8579);
        double r8581 = atan(r8580);
        double r8582 = r8574 * r8581;
        return r8582;
}
↓
double f(double x) {
        double r8583 = 2.0;
        double r8584 = 1.0;
        double r8585 = x;
        double r8586 = r8584 - r8585;
        double r8587 = sqrt(r8586);
        double r8588 = r8584 + r8585;
        double r8589 = sqrt(r8588);
        double r8590 = r8587 / r8589;
        double r8591 = r8590 * r8590;
        double r8592 = sqrt(r8591);
        double r8593 = atan(r8592);
        double r8594 = r8583 * r8593;
        return r8594;
}

Error

Bits error versus x

Try it out

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-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 add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))