Average Error: 13.6 → 14.2
Time: 5.9s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \sqrt[3]{{\left(\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\left(\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}\right)}\]
\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 + \sqrt[3]{{\left(\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \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)}^{3}}\right)}
double f(double p, double x) {
        double r424094 = 0.5;
        double r424095 = 1.0;
        double r424096 = x;
        double r424097 = 4.0;
        double r424098 = p;
        double r424099 = r424097 * r424098;
        double r424100 = r424099 * r424098;
        double r424101 = r424096 * r424096;
        double r424102 = r424100 + r424101;
        double r424103 = sqrt(r424102);
        double r424104 = r424096 / r424103;
        double r424105 = r424095 + r424104;
        double r424106 = r424094 * r424105;
        double r424107 = sqrt(r424106);
        return r424107;
}
↓
double f(double p, double x) {
        double r424108 = 0.5;
        double r424109 = 1.0;
        double r424110 = x;
        double r424111 = 4.0;
        double r424112 = p;
        double r424113 = r424111 * r424112;
        double r424114 = r424113 * r424112;
        double r424115 = r424110 * r424110;
        double r424116 = r424114 + r424115;
        double r424117 = sqrt(r424116);
        double r424118 = sqrt(r424117);
        double r424119 = cbrt(r424117);
        double r424120 = r424119 * r424119;
        double r424121 = r424120 * r424119;
        double r424122 = sqrt(r424121);
        double r424123 = r424118 * r424122;
        double r424124 = r424110 / r424123;
        double r424125 = 3.0;
        double r424126 = pow(r424124, r424125);
        double r424127 = cbrt(r424126);
        double r424128 = r424109 + r424127;
        double r424129 = r424108 * r424128;
        double r424130 = sqrt(r424129);
        return r424130;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.6
Herbie14.2
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.6

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

  3. Applied add-cbrt-cube19.5

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

  4. Applied add-cbrt-cube22.5

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

  5. Applied cbrt-undiv22.5

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

  6. Simplified13.6

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

  8. Applied add-sqr-sqrt13.6

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

  9. Applied sqrt-prod13.7

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

  11. Applied add-cube-cbrt14.2

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

  12. Final simplification14.2

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

Reproduce

herbie shell --seed 2020046 
(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))))))))