Average Error: 13.4 → 14.8
Time: 6.2s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{{\left(1 + \frac{\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}{\sqrt{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \sqrt[3]{{\left(1 + \frac{\frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}{\sqrt{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}^{3}}}
double f(double p, double x) {
        double r209058 = 0.5;
        double r209059 = 1.0;
        double r209060 = x;
        double r209061 = 4.0;
        double r209062 = p;
        double r209063 = r209061 * r209062;
        double r209064 = r209063 * r209062;
        double r209065 = r209060 * r209060;
        double r209066 = r209064 + r209065;
        double r209067 = sqrt(r209066);
        double r209068 = r209060 / r209067;
        double r209069 = r209059 + r209068;
        double r209070 = r209058 * r209069;
        double r209071 = sqrt(r209070);
        return r209071;
}
↓
double f(double p, double x) {
        double r209072 = 0.5;
        double r209073 = 1.0;
        double r209074 = x;
        double r209075 = 4.0;
        double r209076 = p;
        double r209077 = r209075 * r209076;
        double r209078 = r209077 * r209076;
        double r209079 = r209074 * r209074;
        double r209080 = r209078 + r209079;
        double r209081 = sqrt(r209080);
        double r209082 = sqrt(r209081);
        double r209083 = r209074 / r209082;
        double r209084 = cbrt(r209080);
        double r209085 = fabs(r209084);
        double r209086 = sqrt(r209084);
        double r209087 = r209085 * r209086;
        double r209088 = sqrt(r209087);
        double r209089 = r209083 / r209088;
        double r209090 = r209073 + r209089;
        double r209091 = 3.0;
        double r209092 = pow(r209090, r209091);
        double r209093 = cbrt(r209092);
        double r209094 = r209072 * r209093;
        double r209095 = sqrt(r209094);
        return r209095;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target13.4
Herbie14.8
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.4

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

  3. Applied add-sqr-sqrt13.4

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

  4. Applied sqrt-prod14.6

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

  5. Applied associate-/r*14.6

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

  7. Applied add-cube-cbrt14.8

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

  8. Applied sqrt-prod14.8

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

  9. Simplified14.8

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

  11. Applied add-cbrt-cube14.8

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

  12. Simplified14.8

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

  13. Final simplification14.8

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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))))))))