Average Error: 13.3 → 13.7
Time: 24.2s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\frac{\sqrt{\left(0.5 \cdot 0.5\right) \cdot \left(\left(\frac{x \cdot x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot \frac{x}{p \cdot \left(p \cdot 4\right) + x \cdot x}\right) \cdot 0.5 + 0.5\right)}}{\sqrt{\left(\frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot 0.5\right) \cdot \left(\frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot 0.5\right) + \left(0.5 \cdot 0.5 - 0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot 0.5\right)\right)}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\frac{\sqrt{\left(0.5 \cdot 0.5\right) \cdot \left(\left(\frac{x \cdot x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot \frac{x}{p \cdot \left(p \cdot 4\right) + x \cdot x}\right) \cdot 0.5 + 0.5\right)}}{\sqrt{\left(\frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot 0.5\right) \cdot \left(\frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot 0.5\right) + \left(0.5 \cdot 0.5 - 0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot 0.5\right)\right)}}
double f(double p, double x) {
        double r80197220 = 0.5;
        double r80197221 = 1.0;
        double r80197222 = x;
        double r80197223 = 4.0;
        double r80197224 = p;
        double r80197225 = r80197223 * r80197224;
        double r80197226 = r80197225 * r80197224;
        double r80197227 = r80197222 * r80197222;
        double r80197228 = r80197226 + r80197227;
        double r80197229 = sqrt(r80197228);
        double r80197230 = r80197222 / r80197229;
        double r80197231 = r80197221 + r80197230;
        double r80197232 = r80197220 * r80197231;
        double r80197233 = sqrt(r80197232);
        return r80197233;
}
↓
double f(double p, double x) {
        double r80197234 = 0.5;
        double r80197235 = r80197234 * r80197234;
        double r80197236 = x;
        double r80197237 = r80197236 * r80197236;
        double r80197238 = p;
        double r80197239 = 4.0;
        double r80197240 = r80197238 * r80197239;
        double r80197241 = r80197238 * r80197240;
        double r80197242 = r80197241 + r80197237;
        double r80197243 = sqrt(r80197242);
        double r80197244 = r80197237 / r80197243;
        double r80197245 = r80197236 / r80197242;
        double r80197246 = r80197244 * r80197245;
        double r80197247 = r80197246 * r80197234;
        double r80197248 = r80197247 + r80197234;
        double r80197249 = r80197235 * r80197248;
        double r80197250 = sqrt(r80197249);
        double r80197251 = r80197236 / r80197243;
        double r80197252 = r80197251 * r80197234;
        double r80197253 = r80197252 * r80197252;
        double r80197254 = r80197234 * r80197252;
        double r80197255 = r80197235 - r80197254;
        double r80197256 = r80197253 + r80197255;
        double r80197257 = sqrt(r80197256);
        double r80197258 = r80197250 / r80197257;
        return r80197258;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target13.3
Herbie13.7
\[\sqrt{\frac{1}{2} + \frac{\mathsf{copysign}\left(\frac{1}{2}, x\right)}{\mathsf{hypot}\left(1, \left(\frac{2 \cdot p}{x}\right)\right)}}\]

Derivation

  1. Initial program 13.3

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

  2. Simplified13.3

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

  4. Applied flip3-+13.3

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

  5. Applied sqrt-div13.3

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

  6. Simplified13.7

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

  7. Final simplification13.7

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

Reproduce

herbie shell --seed 2019120 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))