Average Error: 13.1 → 13.1
Time: 42.8s
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 r9197384 = 0.5;
        double r9197385 = 1.0;
        double r9197386 = x;
        double r9197387 = 4.0;
        double r9197388 = p;
        double r9197389 = r9197387 * r9197388;
        double r9197390 = r9197389 * r9197388;
        double r9197391 = r9197386 * r9197386;
        double r9197392 = r9197390 + r9197391;
        double r9197393 = sqrt(r9197392);
        double r9197394 = r9197386 / r9197393;
        double r9197395 = r9197385 + r9197394;
        double r9197396 = r9197384 * r9197395;
        double r9197397 = sqrt(r9197396);
        return r9197397;
}
↓
double f(double p, double x) {
        double r9197398 = x;
        double r9197399 = r9197398 * r9197398;
        double r9197400 = p;
        double r9197401 = 4.0;
        double r9197402 = r9197400 * r9197401;
        double r9197403 = r9197402 * r9197400;
        double r9197404 = r9197399 + r9197403;
        double r9197405 = sqrt(r9197404);
        double r9197406 = 0.5;
        double r9197407 = r9197405 / r9197406;
        double r9197408 = r9197398 / r9197407;
        double r9197409 = r9197408 + r9197406;
        double r9197410 = exp(r9197409);
        double r9197411 = log(r9197410);
        double r9197412 = cbrt(r9197411);
        double r9197413 = r9197412 * r9197412;
        double r9197414 = r9197413 * r9197412;
        double r9197415 = cbrt(r9197414);
        double r9197416 = r9197413 * r9197415;
        double r9197417 = sqrt(r9197416);
        return r9197417;
}

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))))))))