Average Error: 13.1 → 14.2
Time: 14.1s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\frac{\left(\frac{0.5}{\frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x} \cdot \frac{\sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}}{x \cdot x}} + 0.5\right) \cdot \left(0.5 \cdot 0.5\right)}{\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} \cdot \left(\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} - 0.5\right) + 0.5 \cdot 0.5}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\frac{\left(\frac{0.5}{\frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x} \cdot \frac{\sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}}{x \cdot x}} + 0.5\right) \cdot \left(0.5 \cdot 0.5\right)}{\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} \cdot \left(\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} - 0.5\right) + 0.5 \cdot 0.5}}
double f(double p, double x) {
        double r6065922 = 0.5;
        double r6065923 = 1.0;
        double r6065924 = x;
        double r6065925 = 4.0;
        double r6065926 = p;
        double r6065927 = r6065925 * r6065926;
        double r6065928 = r6065927 * r6065926;
        double r6065929 = r6065924 * r6065924;
        double r6065930 = r6065928 + r6065929;
        double r6065931 = sqrt(r6065930);
        double r6065932 = r6065924 / r6065931;
        double r6065933 = r6065923 + r6065932;
        double r6065934 = r6065922 * r6065933;
        double r6065935 = sqrt(r6065934);
        return r6065935;
}
↓
double f(double p, double x) {
        double r6065936 = 0.5;
        double r6065937 = p;
        double r6065938 = 4.0;
        double r6065939 = r6065937 * r6065938;
        double r6065940 = r6065939 * r6065937;
        double r6065941 = x;
        double r6065942 = r6065941 * r6065941;
        double r6065943 = r6065940 + r6065942;
        double r6065944 = r6065943 / r6065941;
        double r6065945 = sqrt(r6065943);
        double r6065946 = sqrt(r6065945);
        double r6065947 = r6065946 * r6065946;
        double r6065948 = r6065947 / r6065942;
        double r6065949 = r6065944 * r6065948;
        double r6065950 = r6065936 / r6065949;
        double r6065951 = r6065950 + r6065936;
        double r6065952 = r6065936 * r6065936;
        double r6065953 = r6065951 * r6065952;
        double r6065954 = r6065945 / r6065936;
        double r6065955 = r6065941 / r6065954;
        double r6065956 = r6065955 - r6065936;
        double r6065957 = r6065955 * r6065956;
        double r6065958 = r6065957 + r6065952;
        double r6065959 = r6065953 / r6065958;
        double r6065960 = sqrt(r6065959);
        return r6065960;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.1
Target13.1
Herbie14.2
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \left(\frac{2 \cdot p}{x}\right)\right)}}\]

Derivation

  1. Initial program 13.1

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]

  2. Simplified13.1

    \[\leadsto \color{blue}{\sqrt{0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}} + 0.5}}\]
  3. Using strategy rm

  4. Applied flip3-+13.1

    \[\leadsto \sqrt{\color{blue}{\frac{{\left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right)}^{3} + {0.5}^{3}}{\left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) + \left(0.5 \cdot 0.5 - \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot 0.5\right)}}}\]

  5. Simplified13.7

    \[\leadsto \sqrt{\frac{\color{blue}{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{0.5}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{x \cdot x} \cdot \frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x}} + 0.5\right)}}{\left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) + \left(0.5 \cdot 0.5 - \left(0.5 \cdot \frac{x}{\sqrt{x \cdot x + p \cdot \left(4 \cdot p\right)}}\right) \cdot 0.5\right)}}\]

  6. Simplified13.7

    \[\leadsto \sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{0.5}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{x \cdot x} \cdot \frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x}} + 0.5\right)}{\color{blue}{0.5 \cdot 0.5 + \frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} \cdot \left(\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} - 0.5\right)}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt13.7

    \[\leadsto \sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{0.5}{\frac{\sqrt{\color{blue}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x} \cdot \sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}}}{x \cdot x} \cdot \frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x}} + 0.5\right)}{0.5 \cdot 0.5 + \frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} \cdot \left(\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} - 0.5\right)}}\]

  9. Applied sqrt-prod14.2

    \[\leadsto \sqrt{\frac{\left(0.5 \cdot 0.5\right) \cdot \left(\frac{0.5}{\frac{\color{blue}{\sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}}}{x \cdot x} \cdot \frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x}} + 0.5\right)}{0.5 \cdot 0.5 + \frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} \cdot \left(\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} - 0.5\right)}}\]

  10. Final simplification14.2

    \[\leadsto \sqrt{\frac{\left(\frac{0.5}{\frac{\left(p \cdot 4\right) \cdot p + x \cdot x}{x} \cdot \frac{\sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}}{x \cdot x}} + 0.5\right) \cdot \left(0.5 \cdot 0.5\right)}{\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} \cdot \left(\frac{x}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{0.5}} - 0.5\right) + 0.5 \cdot 0.5}}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))