Average Error: 13.2 → 13.2
Time: 17.5s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\log \left(e^{0.5 + \frac{x \cdot 0.5}{\sqrt{x \cdot x + 4 \cdot \left(p \cdot p\right)}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\log \left(e^{0.5 + \frac{x \cdot 0.5}{\sqrt{x \cdot x + 4 \cdot \left(p \cdot p\right)}}}\right)}
double f(double p, double x) {
        double r7609969 = 0.5;
        double r7609970 = 1.0;
        double r7609971 = x;
        double r7609972 = 4.0;
        double r7609973 = p;
        double r7609974 = r7609972 * r7609973;
        double r7609975 = r7609974 * r7609973;
        double r7609976 = r7609971 * r7609971;
        double r7609977 = r7609975 + r7609976;
        double r7609978 = sqrt(r7609977);
        double r7609979 = r7609971 / r7609978;
        double r7609980 = r7609970 + r7609979;
        double r7609981 = r7609969 * r7609980;
        double r7609982 = sqrt(r7609981);
        return r7609982;
}
↓
double f(double p, double x) {
        double r7609983 = 0.5;
        double r7609984 = x;
        double r7609985 = r7609984 * r7609983;
        double r7609986 = r7609984 * r7609984;
        double r7609987 = 4.0;
        double r7609988 = p;
        double r7609989 = r7609988 * r7609988;
        double r7609990 = r7609987 * r7609989;
        double r7609991 = r7609986 + r7609990;
        double r7609992 = sqrt(r7609991);
        double r7609993 = r7609985 / r7609992;
        double r7609994 = r7609983 + r7609993;
        double r7609995 = exp(r7609994);
        double r7609996 = log(r7609995);
        double r7609997 = sqrt(r7609996);
        return r7609997;
}

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

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. Simplified13.2

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

  4. Applied div-inv13.4

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

  6. Applied add-log-exp13.4

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

  7. Applied add-log-exp13.3

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

  8. Applied sum-log13.2

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

  9. Simplified13.2

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

  10. Final simplification13.2

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

Reproduce

herbie shell --seed 2019141 
(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))))))))