Average Error: 13.0 → 13.0
Time: 37.2s
Precision: 64
\[10^{-150} \lt \left|x\right| \lt 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}, 0.5\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}, 0.5\right)}
double f(double p, double x) {
        double r5472022 = 0.5;
        double r5472023 = 1.0;
        double r5472024 = x;
        double r5472025 = 4.0;
        double r5472026 = p;
        double r5472027 = r5472025 * r5472026;
        double r5472028 = r5472027 * r5472026;
        double r5472029 = r5472024 * r5472024;
        double r5472030 = r5472028 + r5472029;
        double r5472031 = sqrt(r5472030);
        double r5472032 = r5472024 / r5472031;
        double r5472033 = r5472023 + r5472032;
        double r5472034 = r5472022 * r5472033;
        double r5472035 = sqrt(r5472034);
        return r5472035;
}
↓
double f(double p, double x) {
        double r5472036 = 0.5;
        double r5472037 = x;
        double r5472038 = p;
        double r5472039 = 4.0;
        double r5472040 = r5472038 * r5472039;
        double r5472041 = r5472038 * r5472040;
        double r5472042 = fma(r5472037, r5472037, r5472041);
        double r5472043 = sqrt(r5472042);
        double r5472044 = r5472037 / r5472043;
        double r5472045 = fma(r5472036, r5472044, r5472036);
        double r5472046 = sqrt(r5472045);
        return r5472046;
}

Error

Bits error versus p

Bits error versus x

Target

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

Derivation

  1. Initial program 13.0

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

  3. Applied add-log-exp13.0

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

  4. Simplified13.0

    \[\leadsto \log \color{blue}{\left(e^{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5\right)}}\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt13.9

    \[\leadsto \log \left(e^{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}}, 0.5\right)}}\right)\]
  7. Using strategy rm

  8. Applied *-un-lft-identity13.9

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

  9. Applied log-prod13.9

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

  10. Simplified13.9

    \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}, 0.5\right)}}\right)\]

  11. Simplified13.0

    \[\leadsto 0 + \color{blue}{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}, 0.5\right)}}\]

  12. Final simplification13.0

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}, 0.5\right)}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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))))))))