arccos

Percentage Accurate: 100.0% → 100.0%
Time: 6.5s
Alternatives: 7
Speedup: TODO×

Specification

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

Your Program's Arguments

Results

Enter valid numbers for all inputs

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 7 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Alternative 1?

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

  1. Initial program 100.0%

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
  2. Step-by-step derivation

    1. flip-+100.0%

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

    2. associate-/r/100.0%

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

    3. metadata-eval100.0%

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

  3. Applied egg-rr100.0%

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

  4. Taylor expanded in x around 0 100.0%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left(\sqrt{\frac{1}{1 - {x}^{2}}} \cdot \left(1 - x\right)\right)} \]
  5. Step-by-step derivation

    1. *-commutative100.0%

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

    2. unpow2100.0%

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

  6. Simplified100.0%

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

  7. Final simplification100.0%

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

Alternative 2?

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

  1. Initial program 100.0%

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
  2. Step-by-step derivation

    1. div-sub100.0%

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

    2. flip-+100.0%

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

    3. associate-/r/100.0%

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

    4. fma-neg100.0%

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

    5. metadata-eval100.0%

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

  3. Applied egg-rr100.0%

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\frac{x}{1 + x}\right)}}\right) \]
  4. Step-by-step derivation

    1. fma-udef100.0%

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

    2. unsub-neg100.0%

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

    3. associate-*l/100.0%

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

    4. *-un-lft-identity100.0%

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

    5. div-inv100.0%

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

    6. flip-+100.0%

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

    7. metadata-eval100.0%

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

    8. clear-num100.0%

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

    9. *-un-lft-identity100.0%

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

    10. distribute-rgt-out--100.0%

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

    11. flip--100.0%

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

    12. metadata-eval100.0%

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

    13. associate-*r/100.0%

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

  5. Applied egg-rr100.0%

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

  6. Taylor expanded in x around 0 100.0%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left(\frac{1}{1 + x} \cdot \sqrt{1 - {x}^{2}}\right)} \]
  7. Step-by-step derivation

    1. *-commutative100.0%

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

    2. associate-*r/100.0%

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

    3. *-rgt-identity100.0%

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

    4. unpow2100.0%

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

  8. Simplified100.0%

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

  9. Final simplification100.0%

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

Alternative 3?

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

  1. Initial program 100.0%

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

  2. Final simplification100.0%

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

Alternative 4?

\[2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right) \]
Derivation

  1. Initial program 100.0%

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
  2. Step-by-step derivation

    1. flip-+100.0%

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

    2. associate-/r/100.0%

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

    3. metadata-eval100.0%

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

  3. Applied egg-rr100.0%

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

  4. Taylor expanded in x around 0 100.0%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left(\sqrt{\frac{1}{1 - {x}^{2}}} \cdot \left(1 - x\right)\right)} \]
  5. Step-by-step derivation

    1. *-commutative100.0%

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

    2. unpow2100.0%

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

  6. Simplified100.0%

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

  7. Taylor expanded in x around 0 99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)}\right) \]
  8. Step-by-step derivation

    1. *-commutative99.5%

      \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.5}\right)\right) \]

    2. unpow299.5%

      \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5\right)\right) \]

  9. Simplified99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot 0.5\right)}\right) \]

  10. Final simplification99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right) \]

Alternative 5?

\[2 \cdot \tan^{-1} \left(\frac{1 + \left(x \cdot x\right) \cdot -0.5}{1 + x}\right) \]
Derivation

  1. Initial program 100.0%

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
  2. Step-by-step derivation

    1. div-sub100.0%

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

    2. flip-+100.0%

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

    3. associate-/r/100.0%

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

    4. fma-neg100.0%

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

    5. metadata-eval100.0%

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

  3. Applied egg-rr100.0%

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{1 - x \cdot x}, 1 - x, -\frac{x}{1 + x}\right)}}\right) \]
  4. Step-by-step derivation

    1. fma-udef100.0%

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

    2. unsub-neg100.0%

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

    3. associate-*l/100.0%

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

    4. *-un-lft-identity100.0%

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

    5. div-inv100.0%

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

    6. flip-+100.0%

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

    7. metadata-eval100.0%

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

    8. clear-num100.0%

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

    9. *-un-lft-identity100.0%

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

    10. distribute-rgt-out--100.0%

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

    11. flip--100.0%

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

    12. metadata-eval100.0%

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

    13. associate-*r/100.0%

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

  5. Applied egg-rr100.0%

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

  6. Taylor expanded in x around 0 100.0%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left(\frac{1}{1 + x} \cdot \sqrt{1 - {x}^{2}}\right)} \]
  7. Step-by-step derivation

    1. *-commutative100.0%

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

    2. associate-*r/100.0%

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

    3. *-rgt-identity100.0%

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

    4. unpow2100.0%

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

  8. Simplified100.0%

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

  9. Taylor expanded in x around 0 99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\frac{\color{blue}{1 + -0.5 \cdot {x}^{2}}}{1 + x}\right) \]
  10. Step-by-step derivation

    1. unpow299.5%

      \[\leadsto 2 \cdot \tan^{-1} \left(\frac{1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{1 + x}\right) \]

  11. Simplified99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\frac{\color{blue}{1 + -0.5 \cdot \left(x \cdot x\right)}}{1 + x}\right) \]

  12. Final simplification99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\frac{1 + \left(x \cdot x\right) \cdot -0.5}{1 + x}\right) \]

Alternative 6?

\[2 \cdot \tan^{-1} \left(1 - x\right) \]
Derivation

  1. Initial program 100.0%

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

  2. Taylor expanded in x around 0 99.1%

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{1 + -2 \cdot x}}\right) \]
  3. Step-by-step derivation

    1. *-commutative99.1%

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

  4. Simplified99.1%

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

  5. Taylor expanded in x around 0 99.1%

    \[\leadsto 2 \cdot \tan^{-1} \color{blue}{\left(1 + -1 \cdot x\right)} \]
  6. Step-by-step derivation

    1. neg-mul-199.1%

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

    2. sub-neg99.1%

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

  7. Simplified99.1%

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

  8. Final simplification99.1%

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

Alternative 7?

\[2 \cdot \tan^{-1} 1 \]
Derivation

  1. Initial program 100.0%

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \]
  2. Step-by-step derivation

    1. flip-+100.0%

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

    2. associate-/r/100.0%

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

    3. metadata-eval100.0%

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

  3. Applied egg-rr100.0%

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

  4. Taylor expanded in x around 0 100.0%

    \[\leadsto 2 \cdot \color{blue}{\tan^{-1} \left(\sqrt{\frac{1}{1 - {x}^{2}}} \cdot \left(1 - x\right)\right)} \]
  5. Step-by-step derivation

    1. *-commutative100.0%

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

    2. unpow2100.0%

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

  6. Simplified100.0%

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

  7. Taylor expanded in x around 0 99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \color{blue}{\left(1 + 0.5 \cdot {x}^{2}\right)}\right) \]
  8. Step-by-step derivation

    1. *-commutative99.5%

      \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.5}\right)\right) \]

    2. unpow299.5%

      \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.5\right)\right) \]

  9. Simplified99.5%

    \[\leadsto 2 \cdot \tan^{-1} \left(\left(1 - x\right) \cdot \color{blue}{\left(1 + \left(x \cdot x\right) \cdot 0.5\right)}\right) \]

  10. Taylor expanded in x around 0 97.9%

    \[\leadsto 2 \cdot \tan^{-1} \color{blue}{1} \]

  11. Final simplification97.9%

    \[\leadsto 2 \cdot \tan^{-1} 1 \]

Reproduce

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herbie shell --seed 2023166 
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2.0 (atan (sqrt (/ (- 1.0 x) (+ 1.0 x))))))