Average Error: 13.0 → 13.0
Time: 4.5s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{\sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}}{\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 \frac{\sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}
double f(double p, double x) {
        double r330188 = 0.5;
        double r330189 = 1.0;
        double r330190 = x;
        double r330191 = 4.0;
        double r330192 = p;
        double r330193 = r330191 * r330192;
        double r330194 = r330193 * r330192;
        double r330195 = r330190 * r330190;
        double r330196 = r330194 + r330195;
        double r330197 = sqrt(r330196);
        double r330198 = r330190 / r330197;
        double r330199 = r330189 + r330198;
        double r330200 = r330188 * r330199;
        double r330201 = sqrt(r330200);
        return r330201;
}
↓
double f(double p, double x) {
        double r330202 = 0.5;
        double r330203 = 1.0;
        double r330204 = 3.0;
        double r330205 = pow(r330203, r330204);
        double r330206 = x;
        double r330207 = 4.0;
        double r330208 = p;
        double r330209 = r330207 * r330208;
        double r330210 = r330209 * r330208;
        double r330211 = r330206 * r330206;
        double r330212 = r330210 + r330211;
        double r330213 = sqrt(r330212);
        double r330214 = r330206 / r330213;
        double r330215 = pow(r330214, r330204);
        double r330216 = r330205 + r330215;
        double r330217 = r330203 - r330214;
        double r330218 = r330203 * r330217;
        double r330219 = 2.0;
        double r330220 = pow(r330206, r330219);
        double r330221 = cbrt(r330220);
        double r330222 = r330221 * r330221;
        double r330223 = cbrt(r330212);
        double r330224 = r330223 * r330223;
        double r330225 = r330222 / r330224;
        double r330226 = r330221 / r330223;
        double r330227 = r330225 * r330226;
        double r330228 = r330218 + r330227;
        double r330229 = r330216 / r330228;
        double r330230 = r330202 * r330229;
        double r330231 = sqrt(r330230);
        return r330231;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.0
Target13.0
Herbie13.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  3. Applied flip3-+13.0

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

  4. Simplified13.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\color{blue}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt13.2

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{{x}^{2}}{\color{blue}{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\]

  7. Applied add-cube-cbrt13.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{\color{blue}{\left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \sqrt[3]{{x}^{2}}}}{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\]

  8. Applied times-frac13.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \color{blue}{\frac{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{\sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\]

  9. Final simplification13.0

    \[\leadsto \sqrt{0.5 \cdot \frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot \left(1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) + \frac{\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{\sqrt[3]{{x}^{2}}}{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\]

Reproduce

herbie shell --seed 2020059 
(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))))))))