Average Error: 13.2 → 14.1
Time: 5.4s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{\frac{\sqrt[3]{x}}{\frac{\left|\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right|}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}{\sqrt{\left(\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \frac{\frac{\sqrt[3]{x}}{\frac{\left|\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right|}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\sqrt{\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}{\sqrt{\left(\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}
double f(double p, double x) {
        double r377423 = 0.5;
        double r377424 = 1.0;
        double r377425 = x;
        double r377426 = 4.0;
        double r377427 = p;
        double r377428 = r377426 * r377427;
        double r377429 = r377428 * r377427;
        double r377430 = r377425 * r377425;
        double r377431 = r377429 + r377430;
        double r377432 = sqrt(r377431);
        double r377433 = r377425 / r377432;
        double r377434 = r377424 + r377433;
        double r377435 = r377423 * r377434;
        double r377436 = sqrt(r377435);
        return r377436;
}
↓
double f(double p, double x) {
        double r377437 = 0.5;
        double r377438 = 1.0;
        double r377439 = x;
        double r377440 = cbrt(r377439);
        double r377441 = 4.0;
        double r377442 = p;
        double r377443 = r377441 * r377442;
        double r377444 = r377443 * r377442;
        double r377445 = r377439 * r377439;
        double r377446 = r377444 + r377445;
        double r377447 = sqrt(r377446);
        double r377448 = cbrt(r377447);
        double r377449 = fabs(r377448);
        double r377450 = r377449 / r377440;
        double r377451 = r377440 / r377450;
        double r377452 = sqrt(r377448);
        double r377453 = r377440 / r377452;
        double r377454 = r377451 * r377453;
        double r377455 = r377448 * r377448;
        double r377456 = r377455 * r377448;
        double r377457 = sqrt(r377456);
        double r377458 = r377454 / r377457;
        double r377459 = r377438 + r377458;
        double r377460 = r377437 * r377459;
        double r377461 = sqrt(r377460);
        return r377461;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target13.2
Herbie14.1
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.2

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

  3. Applied add-sqr-sqrt13.2

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

  4. Applied sqrt-prod14.3

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

  5. Applied associate-/r*14.3

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

  7. Applied add-cube-cbrt14.7

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

  8. Applied sqrt-prod14.7

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

  9. Applied add-cube-cbrt14.7

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

  10. Applied times-frac14.7

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

  11. Simplified14.7

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

  13. Applied add-cube-cbrt14.1

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

  14. Final simplification14.1

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

Reproduce

herbie shell --seed 2020033 
(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))))))))