Average Error: 13.3 → 13.8
Time: 5.3s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\left(\sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\left(\sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}
double f(double p, double x) {
        double r391013 = 0.5;
        double r391014 = 1.0;
        double r391015 = x;
        double r391016 = 4.0;
        double r391017 = p;
        double r391018 = r391016 * r391017;
        double r391019 = r391018 * r391017;
        double r391020 = r391015 * r391015;
        double r391021 = r391019 + r391020;
        double r391022 = sqrt(r391021);
        double r391023 = r391015 / r391022;
        double r391024 = r391014 + r391023;
        double r391025 = r391013 * r391024;
        double r391026 = sqrt(r391025);
        return r391026;
}
↓
double f(double p, double x) {
        double r391027 = 0.5;
        double r391028 = 1.0;
        double r391029 = x;
        double r391030 = 4.0;
        double r391031 = p;
        double r391032 = r391030 * r391031;
        double r391033 = r391032 * r391031;
        double r391034 = r391029 * r391029;
        double r391035 = r391033 + r391034;
        double r391036 = sqrt(r391035);
        double r391037 = r391029 / r391036;
        double r391038 = r391028 + r391037;
        double r391039 = r391027 * r391038;
        double r391040 = sqrt(r391039);
        double r391041 = cbrt(r391040);
        double r391042 = r391041 * r391041;
        double r391043 = cbrt(r391038);
        double r391044 = r391043 * r391043;
        double r391045 = r391044 * r391043;
        double r391046 = r391027 * r391045;
        double r391047 = sqrt(r391046);
        double r391048 = cbrt(r391047);
        double r391049 = r391042 * r391048;
        return r391049;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target13.3
Herbie13.8
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt13.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt13.8

    \[\leadsto \left(\sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \cdot \sqrt[3]{\sqrt{0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}}\]

  6. Final simplification13.8

    \[\leadsto \left(\sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right) \cdot \sqrt[3]{\sqrt{0.5 \cdot \left(\left(\sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1 (/ (* 2 p) x)))))

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