Average Error: 59.4 → 0.2
Time: 15.9s
Precision: 64
\[\frac{1 - \sqrt{1 - {t}^{2}}}{t}\]
\[\left(\left(\frac{\frac{1}{4} \cdot {t}^{5}}{{\left(\sqrt{1} + 1\right)}^{3} \cdot 1} + \frac{1}{8} \cdot \frac{{t}^{5}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot {\left(\sqrt{1}\right)}^{3}}\right) + \frac{t}{\sqrt{1} + 1}\right) + \frac{1}{2} \cdot \frac{{t}^{3}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot \sqrt{1}}\]
\frac{1 - \sqrt{1 - {t}^{2}}}{t}
\left(\left(\frac{\frac{1}{4} \cdot {t}^{5}}{{\left(\sqrt{1} + 1\right)}^{3} \cdot 1} + \frac{1}{8} \cdot \frac{{t}^{5}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot {\left(\sqrt{1}\right)}^{3}}\right) + \frac{t}{\sqrt{1} + 1}\right) + \frac{1}{2} \cdot \frac{{t}^{3}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot \sqrt{1}}
double f(double t) {
        double r125030 = 1.0;
        double r125031 = t;
        double r125032 = 2.0;
        double r125033 = pow(r125031, r125032);
        double r125034 = r125030 - r125033;
        double r125035 = sqrt(r125034);
        double r125036 = r125030 - r125035;
        double r125037 = r125036 / r125031;
        return r125037;
}

double f(double t) {
        double r125038 = 0.25;
        double r125039 = t;
        double r125040 = 5.0;
        double r125041 = pow(r125039, r125040);
        double r125042 = r125038 * r125041;
        double r125043 = 1.0;
        double r125044 = sqrt(r125043);
        double r125045 = r125044 + r125043;
        double r125046 = 3.0;
        double r125047 = pow(r125045, r125046);
        double r125048 = r125047 * r125043;
        double r125049 = r125042 / r125048;
        double r125050 = 0.125;
        double r125051 = 2.0;
        double r125052 = pow(r125045, r125051);
        double r125053 = pow(r125044, r125046);
        double r125054 = r125052 * r125053;
        double r125055 = r125041 / r125054;
        double r125056 = r125050 * r125055;
        double r125057 = r125049 + r125056;
        double r125058 = r125039 / r125045;
        double r125059 = r125057 + r125058;
        double r125060 = 0.5;
        double r125061 = pow(r125039, r125046);
        double r125062 = r125052 * r125044;
        double r125063 = r125061 / r125062;
        double r125064 = r125060 * r125063;
        double r125065 = r125059 + r125064;
        return r125065;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.4

    \[\frac{1 - \sqrt{1 - {t}^{2}}}{t}\]
  2. Using strategy rm
  3. Applied flip--59.4

    \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \sqrt{1 - {t}^{2}} \cdot \sqrt{1 - {t}^{2}}}{1 + \sqrt{1 - {t}^{2}}}}}{t}\]
  4. Simplified28.7

    \[\leadsto \frac{\frac{\color{blue}{\left(1 \cdot 1 - 1\right) + {t}^{2}}}{1 + \sqrt{1 - {t}^{2}}}}{t}\]
  5. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{t}^{3}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot \sqrt{1}} + \left(\frac{1}{4} \cdot \frac{{t}^{5}}{{\left(\sqrt{1} + 1\right)}^{3} \cdot {\left(\sqrt{1}\right)}^{2}} + \left(\frac{1}{8} \cdot \frac{{t}^{5}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot {\left(\sqrt{1}\right)}^{3}} + \frac{t}{\sqrt{1} + 1}\right)\right)}\]
  6. Simplified0.2

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{1}{4} \cdot {t}^{5}}{{\left(\sqrt{1} + 1\right)}^{3} \cdot 1} + \frac{1}{8} \cdot \frac{{t}^{5}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot {\left(\sqrt{1}\right)}^{3}}\right) + \frac{t}{\sqrt{1} + 1}\right) + \frac{1}{2} \cdot \frac{{t}^{3}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot \sqrt{1}}}\]
  7. Final simplification0.2

    \[\leadsto \left(\left(\frac{\frac{1}{4} \cdot {t}^{5}}{{\left(\sqrt{1} + 1\right)}^{3} \cdot 1} + \frac{1}{8} \cdot \frac{{t}^{5}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot {\left(\sqrt{1}\right)}^{3}}\right) + \frac{t}{\sqrt{1} + 1}\right) + \frac{1}{2} \cdot \frac{{t}^{3}}{{\left(\sqrt{1} + 1\right)}^{2} \cdot \sqrt{1}}\]

Reproduce

herbie shell --seed 1 
(FPCore (t)
  :name "(1 - sqrt(1 - t^2))/t"
  :precision binary64
  (/ (- 1 (sqrt (- 1 (pow t 2)))) t))