Average Error: 13.1 → 13.1
Time: 43.8s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(\sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(\sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)} \cdot \sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}\right)}}}
double f(double p, double x) {
        double r8947926 = 0.5;
        double r8947927 = 1.0;
        double r8947928 = x;
        double r8947929 = 4.0;
        double r8947930 = p;
        double r8947931 = r8947929 * r8947930;
        double r8947932 = r8947931 * r8947930;
        double r8947933 = r8947928 * r8947928;
        double r8947934 = r8947932 + r8947933;
        double r8947935 = sqrt(r8947934);
        double r8947936 = r8947928 / r8947935;
        double r8947937 = r8947927 + r8947936;
        double r8947938 = r8947926 * r8947937;
        double r8947939 = sqrt(r8947938);
        return r8947939;
}
↓
double f(double p, double x) {
        double r8947940 = x;
        double r8947941 = r8947940 * r8947940;
        double r8947942 = p;
        double r8947943 = 4.0;
        double r8947944 = r8947942 * r8947943;
        double r8947945 = r8947944 * r8947942;
        double r8947946 = r8947941 + r8947945;
        double r8947947 = sqrt(r8947946);
        double r8947948 = 0.5;
        double r8947949 = r8947947 / r8947948;
        double r8947950 = r8947940 / r8947949;
        double r8947951 = r8947950 + r8947948;
        double r8947952 = exp(r8947951);
        double r8947953 = log(r8947952);
        double r8947954 = cbrt(r8947953);
        double r8947955 = r8947954 * r8947954;
        double r8947956 = r8947955 * r8947954;
        double r8947957 = cbrt(r8947956);
        double r8947958 = r8947955 * r8947957;
        double r8947959 = sqrt(r8947958);
        return r8947959;
}

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
Herbie13.1
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

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

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

  4. Applied add-log-exp13.1

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

  6. Applied add-cube-cbrt13.1

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

  8. Applied add-cube-cbrt13.1

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

  9. Final simplification13.1

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

Reproduce

herbie shell --seed 2019163 
(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))))))))