Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[\sqrt{1 - x \cdot x}\]
↓
\[\frac{\sqrt{{1}^{3} - {\left(x \cdot x\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}}\]
\sqrt{1 - x \cdot x}
↓
\frac{\sqrt{{1}^{3} - {\left(x \cdot x\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}}
double f(double x) {
        double r148040 = 1.0;
        double r148041 = x;
        double r148042 = r148041 * r148041;
        double r148043 = r148040 - r148042;
        double r148044 = sqrt(r148043);
        return r148044;
}
↓
double f(double x) {
        double r148045 = 1.0;
        double r148046 = 3.0;
        double r148047 = pow(r148045, r148046);
        double r148048 = x;
        double r148049 = r148048 * r148048;
        double r148050 = pow(r148049, r148046);
        double r148051 = r148047 - r148050;
        double r148052 = sqrt(r148051);
        double r148053 = r148045 * r148045;
        double r148054 = r148049 * r148049;
        double r148055 = r148045 * r148049;
        double r148056 = r148054 + r148055;
        double r148057 = r148053 + r148056;
        double r148058 = sqrt(r148057);
        double r148059 = r148052 / r148058;
        return r148059;
}

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

    \[\sqrt{1 - x \cdot x}\]
  2. Using strategy rm

  3. Applied flip3--0.0

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

  4. Applied sqrt-div0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Diagrams.TwoD.Ellipse:ellipse from diagrams-lib-1.3.0.3"
  :precision binary64
  (sqrt (- 1 (* x x))))