Average Error: 13.2 → 13.2
Time: 5.2s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \left(\sqrt[3]{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
\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 + \left(\sqrt[3]{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}} \cdot \sqrt[3]{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right) \cdot \sqrt[3]{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}
double f(double p, double x) {
        double r376086 = 0.5;
        double r376087 = 1.0;
        double r376088 = x;
        double r376089 = 4.0;
        double r376090 = p;
        double r376091 = r376089 * r376090;
        double r376092 = r376091 * r376090;
        double r376093 = r376088 * r376088;
        double r376094 = r376092 + r376093;
        double r376095 = sqrt(r376094);
        double r376096 = r376088 / r376095;
        double r376097 = r376087 + r376096;
        double r376098 = r376086 * r376097;
        double r376099 = sqrt(r376098);
        return r376099;
}
↓
double f(double p, double x) {
        double r376100 = 0.5;
        double r376101 = 1.0;
        double r376102 = x;
        double r376103 = 1.0;
        double r376104 = 4.0;
        double r376105 = p;
        double r376106 = r376104 * r376105;
        double r376107 = r376106 * r376105;
        double r376108 = r376102 * r376102;
        double r376109 = r376107 + r376108;
        double r376110 = sqrt(r376109);
        double r376111 = r376103 / r376110;
        double r376112 = r376102 * r376111;
        double r376113 = cbrt(r376112);
        double r376114 = r376113 * r376113;
        double r376115 = r376114 * r376113;
        double r376116 = r376101 + r376115;
        double r376117 = r376100 * r376116;
        double r376118 = sqrt(r376117);
        return r376118;
}

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
Herbie13.2
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.2

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

  3. Applied div-inv13.4

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

  5. Applied add-cube-cbrt13.2

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

  6. Final simplification13.2

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

Reproduce

herbie shell --seed 2020039 +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))))))))