Average Error: 13.3 → 13.3
Time: 47.4s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\log \left(e^{\sqrt[3]{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5} \cdot \sqrt[3]{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}}\right) \cdot \sqrt[3]{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\log \left(e^{\sqrt[3]{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5} \cdot \sqrt[3]{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}}\right) \cdot \sqrt[3]{\frac{x}{\frac{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}}{0.5}} + 0.5}}
double f(double p, double x) {
        double r8492279 = 0.5;
        double r8492280 = 1.0;
        double r8492281 = x;
        double r8492282 = 4.0;
        double r8492283 = p;
        double r8492284 = r8492282 * r8492283;
        double r8492285 = r8492284 * r8492283;
        double r8492286 = r8492281 * r8492281;
        double r8492287 = r8492285 + r8492286;
        double r8492288 = sqrt(r8492287);
        double r8492289 = r8492281 / r8492288;
        double r8492290 = r8492280 + r8492289;
        double r8492291 = r8492279 * r8492290;
        double r8492292 = sqrt(r8492291);
        return r8492292;
}
↓
double f(double p, double x) {
        double r8492293 = x;
        double r8492294 = r8492293 * r8492293;
        double r8492295 = p;
        double r8492296 = 4.0;
        double r8492297 = r8492295 * r8492296;
        double r8492298 = r8492297 * r8492295;
        double r8492299 = r8492294 + r8492298;
        double r8492300 = sqrt(r8492299);
        double r8492301 = 0.5;
        double r8492302 = r8492300 / r8492301;
        double r8492303 = r8492293 / r8492302;
        double r8492304 = r8492303 + r8492301;
        double r8492305 = cbrt(r8492304);
        double r8492306 = r8492305 * r8492305;
        double r8492307 = exp(r8492306);
        double r8492308 = log(r8492307);
        double r8492309 = r8492308 * r8492305;
        double r8492310 = sqrt(r8492309);
        return r8492310;
}

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

Derivation

  1. Initial program 13.3

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

  2. Simplified13.3

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

  4. Applied add-log-exp13.3

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

  6. Applied add-cube-cbrt13.3

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

  7. Applied exp-prod13.3

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

  8. Applied log-pow13.3

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

  9. Final simplification13.3

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

Reproduce

herbie shell --seed 2019168 
(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))))))))