Average Error: 14.0 → 14.0
Time: 20.6s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{\sqrt{\frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} + 0.5} \cdot \frac{0.5 \cdot \left(0.5 \cdot 0.5\right) + \frac{x \cdot 0.5}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{\frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{\left(p \cdot 4\right) \cdot p + x \cdot x}}}}{0.5 \cdot 0.5 + \left(\frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot \frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} - 0.5 \cdot \frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{\sqrt{\frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} + 0.5} \cdot \frac{0.5 \cdot \left(0.5 \cdot 0.5\right) + \frac{x \cdot 0.5}{\frac{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}{\frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot 0.5\right)}{\left(p \cdot 4\right) \cdot p + x \cdot x}}}}{0.5 \cdot 0.5 + \left(\frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot \frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} - 0.5 \cdot \frac{x \cdot 0.5}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}}\right)}}
double f(double p, double x) {
        double r3443046 = 0.5;
        double r3443047 = 1.0;
        double r3443048 = x;
        double r3443049 = 4.0;
        double r3443050 = p;
        double r3443051 = r3443049 * r3443050;
        double r3443052 = r3443051 * r3443050;
        double r3443053 = r3443048 * r3443048;
        double r3443054 = r3443052 + r3443053;
        double r3443055 = sqrt(r3443054);
        double r3443056 = r3443048 / r3443055;
        double r3443057 = r3443047 + r3443056;
        double r3443058 = r3443046 * r3443057;
        double r3443059 = sqrt(r3443058);
        return r3443059;
}
↓
double f(double p, double x) {
        double r3443060 = x;
        double r3443061 = 0.5;
        double r3443062 = r3443060 * r3443061;
        double r3443063 = p;
        double r3443064 = 4.0;
        double r3443065 = r3443063 * r3443064;
        double r3443066 = r3443065 * r3443063;
        double r3443067 = r3443060 * r3443060;
        double r3443068 = r3443066 + r3443067;
        double r3443069 = sqrt(r3443068);
        double r3443070 = r3443062 / r3443069;
        double r3443071 = r3443070 + r3443061;
        double r3443072 = sqrt(r3443071);
        double r3443073 = r3443061 * r3443061;
        double r3443074 = r3443061 * r3443073;
        double r3443075 = r3443062 * r3443062;
        double r3443076 = r3443075 / r3443068;
        double r3443077 = r3443069 / r3443076;
        double r3443078 = r3443062 / r3443077;
        double r3443079 = r3443074 + r3443078;
        double r3443080 = r3443070 * r3443070;
        double r3443081 = r3443061 * r3443070;
        double r3443082 = r3443080 - r3443081;
        double r3443083 = r3443073 + r3443082;
        double r3443084 = r3443079 / r3443083;
        double r3443085 = r3443072 * r3443084;
        double r3443086 = cbrt(r3443085);
        return r3443086;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target14.0
Herbie14.0
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 14.0

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

  2. Simplified14.0

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

  4. Applied add-cbrt-cube14.0

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

  5. Simplified14.0

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

  7. Applied flip3-+14.0

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

  8. Simplified14.0

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

  9. Final simplification14.0

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

Reproduce

herbie shell --seed 2019154 
(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))))))))