Average Error: 12.9 → 12.9
Time: 16.3s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{\log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \frac{\log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}
double f(double p, double x) {
        double r164205 = 0.5;
        double r164206 = 1.0;
        double r164207 = x;
        double r164208 = 4.0;
        double r164209 = p;
        double r164210 = r164208 * r164209;
        double r164211 = r164210 * r164209;
        double r164212 = r164207 * r164207;
        double r164213 = r164211 + r164212;
        double r164214 = sqrt(r164213);
        double r164215 = r164207 / r164214;
        double r164216 = r164206 + r164215;
        double r164217 = r164205 * r164216;
        double r164218 = sqrt(r164217);
        return r164218;
}
↓
double f(double p, double x) {
        double r164219 = 0.5;
        double r164220 = 1.0;
        double r164221 = 3.0;
        double r164222 = pow(r164220, r164221);
        double r164223 = x;
        double r164224 = 4.0;
        double r164225 = p;
        double r164226 = r164224 * r164225;
        double r164227 = r164226 * r164225;
        double r164228 = r164223 * r164223;
        double r164229 = r164227 + r164228;
        double r164230 = sqrt(r164229);
        double r164231 = r164223 / r164230;
        double r164232 = pow(r164231, r164221);
        double r164233 = r164222 + r164232;
        double r164234 = exp(r164233);
        double r164235 = log(r164234);
        double r164236 = r164220 - r164231;
        double r164237 = r164220 * r164236;
        double r164238 = 2.0;
        double r164239 = pow(r164225, r164238);
        double r164240 = r164224 * r164239;
        double r164241 = pow(r164223, r164238);
        double r164242 = r164240 + r164241;
        double r164243 = r164228 / r164242;
        double r164244 = r164237 + r164243;
        double r164245 = r164235 / r164244;
        double r164246 = r164219 * r164245;
        double r164247 = sqrt(r164246);
        return r164247;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.9
Target12.9
Herbie12.9
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.9

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

  3. Applied flip3-+12.9

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

  4. Simplified12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\color{blue}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}}\]
  5. Using strategy rm

  6. Applied add-log-exp12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + \color{blue}{\log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  7. Applied add-log-exp12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(e^{{1}^{3}}\right)} + \log \left(e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  8. Applied sum-log12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\log \left(e^{{1}^{3}} \cdot e^{{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  9. Simplified12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\log \color{blue}{\left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

  10. Final simplification12.9

    \[\leadsto \sqrt{0.5 \cdot \frac{\log \left(e^{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}\right)}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{x \cdot x}{4 \cdot {p}^{2} + {x}^{2}}}}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1.00000000000000001e-150 (fabs x) 9.99999999999999981e149)

  :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))))))))