Average Error: 13.3 → 13.5
Time: 8.8s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\left(\frac{1}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot x + 1\right) \cdot 0.5}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\left(\frac{1}{\sqrt{\left(p \cdot 4\right) \cdot p + x \cdot x}} \cdot x + 1\right) \cdot 0.5}
double f(double p, double x) {
        double r276710 = 0.5;
        double r276711 = 1.0;
        double r276712 = x;
        double r276713 = 4.0;
        double r276714 = p;
        double r276715 = r276713 * r276714;
        double r276716 = r276715 * r276714;
        double r276717 = r276712 * r276712;
        double r276718 = r276716 + r276717;
        double r276719 = sqrt(r276718);
        double r276720 = r276712 / r276719;
        double r276721 = r276711 + r276720;
        double r276722 = r276710 * r276721;
        double r276723 = sqrt(r276722);
        return r276723;
}
↓
double f(double p, double x) {
        double r276724 = 1.0;
        double r276725 = p;
        double r276726 = 4.0;
        double r276727 = r276725 * r276726;
        double r276728 = r276727 * r276725;
        double r276729 = x;
        double r276730 = r276729 * r276729;
        double r276731 = r276728 + r276730;
        double r276732 = sqrt(r276731);
        double r276733 = r276724 / r276732;
        double r276734 = r276733 * r276729;
        double r276735 = 1.0;
        double r276736 = r276734 + r276735;
        double r276737 = 0.5;
        double r276738 = r276736 * r276737;
        double r276739 = sqrt(r276738);
        return r276739;
}

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
Herbie13.5
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

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

  2. Taylor expanded around 0 13.3

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

  3. Simplified13.3

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

  5. Applied div-inv13.5

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

  6. Simplified13.5

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

  7. Final simplification13.5

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

Reproduce

herbie shell --seed 2019194 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))