Average Error: 1.4 → 0.6
Time: 13.4s
Precision: 64
\[\sqrt{1 + \cos x} - \sqrt{1 - \sin x}\]
\[\log \left(e^{\frac{\left(\cos x + 1\right) - \left(1 - \sin x\right)}{\sqrt{1 + \cos x} + \sqrt{1 - \sin x}}}\right)\]
\sqrt{1 + \cos x} - \sqrt{1 - \sin x}
\log \left(e^{\frac{\left(\cos x + 1\right) - \left(1 - \sin x\right)}{\sqrt{1 + \cos x} + \sqrt{1 - \sin x}}}\right)
double f(double x) {
        double r1054880 = 1.0;
        double r1054881 = x;
        double r1054882 = cos(r1054881);
        double r1054883 = r1054880 + r1054882;
        double r1054884 = sqrt(r1054883);
        double r1054885 = sin(r1054881);
        double r1054886 = r1054880 - r1054885;
        double r1054887 = sqrt(r1054886);
        double r1054888 = r1054884 - r1054887;
        return r1054888;
}

double f(double x) {
        double r1054889 = x;
        double r1054890 = cos(r1054889);
        double r1054891 = 1.0;
        double r1054892 = r1054890 + r1054891;
        double r1054893 = sin(r1054889);
        double r1054894 = r1054891 - r1054893;
        double r1054895 = r1054892 - r1054894;
        double r1054896 = r1054891 + r1054890;
        double r1054897 = sqrt(r1054896);
        double r1054898 = sqrt(r1054894);
        double r1054899 = r1054897 + r1054898;
        double r1054900 = r1054895 / r1054899;
        double r1054901 = exp(r1054900);
        double r1054902 = log(r1054901);
        return r1054902;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.4

    \[\sqrt{1 + \cos x} - \sqrt{1 - \sin x}\]
  2. Using strategy rm
  3. Applied flip--1.8

    \[\leadsto \color{blue}{\frac{\sqrt{1 + \cos x} \cdot \sqrt{1 + \cos x} - \sqrt{1 - \sin x} \cdot \sqrt{1 - \sin x}}{\sqrt{1 + \cos x} + \sqrt{1 - \sin x}}}\]
  4. Simplified1.0

    \[\leadsto \frac{\color{blue}{\left(\cos x + 1\right) - \left(1 - \sin x\right)}}{\sqrt{1 + \cos x} + \sqrt{1 - \sin x}}\]
  5. Using strategy rm
  6. Applied add-log-exp0.6

    \[\leadsto \color{blue}{\log \left(e^{\frac{\left(\cos x + 1\right) - \left(1 - \sin x\right)}{\sqrt{1 + \cos x} + \sqrt{1 - \sin x}}}\right)}\]
  7. Final simplification0.6

    \[\leadsto \log \left(e^{\frac{\left(\cos x + 1\right) - \left(1 - \sin x\right)}{\sqrt{1 + \cos x} + \sqrt{1 - \sin x}}}\right)\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "sqrt(1+cos(x)) - sqrt(1-sin(x))"
  :precision binary64
  (- (sqrt (+ 1 (cos x))) (sqrt (- 1 (sin x)))))