Average Error: 0.0 → 0.2
Time: 6.8s
Precision: 64
\[\sqrt{1.0 - x \cdot x}\]
↓
\[\left(\sqrt{1.0} - \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\sqrt{1.0}}\right) - \frac{\frac{\frac{1}{8}}{1.0} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\sqrt{1.0}}\]
\sqrt{1.0 - x \cdot x}
↓
\left(\sqrt{1.0} - \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\sqrt{1.0}}\right) - \frac{\frac{\frac{1}{8}}{1.0} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\sqrt{1.0}}
double f(double x) {
        double r9271205 = 1.0;
        double r9271206 = x;
        double r9271207 = r9271206 * r9271206;
        double r9271208 = r9271205 - r9271207;
        double r9271209 = sqrt(r9271208);
        return r9271209;
}
↓
double f(double x) {
        double r9271210 = 1.0;
        double r9271211 = sqrt(r9271210);
        double r9271212 = x;
        double r9271213 = r9271212 * r9271212;
        double r9271214 = 0.5;
        double r9271215 = r9271213 * r9271214;
        double r9271216 = r9271215 / r9271211;
        double r9271217 = r9271211 - r9271216;
        double r9271218 = 0.125;
        double r9271219 = r9271218 / r9271210;
        double r9271220 = r9271213 * r9271213;
        double r9271221 = r9271219 * r9271220;
        double r9271222 = r9271221 / r9271211;
        double r9271223 = r9271217 - r9271222;
        return r9271223;
}

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

    \[\sqrt{1.0 - x \cdot x}\]

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\sqrt{1.0} - \left(\frac{1}{8} \cdot \frac{{x}^{4}}{{\left(\sqrt{1.0}\right)}^{3}} + \frac{1}{2} \cdot \frac{{x}^{2}}{\sqrt{1.0}}\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\left(\sqrt{1.0} - \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\sqrt{1.0}}\right) - \frac{\frac{\frac{1}{8}}{1.0} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\sqrt{1.0}}}\]

  4. Final simplification0.2

    \[\leadsto \left(\sqrt{1.0} - \frac{\left(x \cdot x\right) \cdot \frac{1}{2}}{\sqrt{1.0}}\right) - \frac{\frac{\frac{1}{8}}{1.0} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\sqrt{1.0}}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  (sqrt (- 1.0 (* x x))))