Average Error: 13.2 → 13.5
Time: 4.5s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \sqrt[3]{{\left(x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}^{3}}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \sqrt[3]{{\left(x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}^{3}}}
double f(double p, double x) {
        double r268068 = 0.5;
        double r268069 = 1.0;
        double r268070 = x;
        double r268071 = 4.0;
        double r268072 = p;
        double r268073 = r268071 * r268072;
        double r268074 = r268073 * r268072;
        double r268075 = r268070 * r268070;
        double r268076 = r268074 + r268075;
        double r268077 = sqrt(r268076);
        double r268078 = r268070 / r268077;
        double r268079 = r268069 + r268078;
        double r268080 = r268068 * r268079;
        double r268081 = sqrt(r268080);
        return r268081;
}
↓
double f(double p, double x) {
        double r268082 = 0.5;
        double r268083 = x;
        double r268084 = 1.0;
        double r268085 = 4.0;
        double r268086 = p;
        double r268087 = r268085 * r268086;
        double r268088 = r268087 * r268086;
        double r268089 = r268083 * r268083;
        double r268090 = r268088 + r268089;
        double r268091 = sqrt(r268090);
        double r268092 = r268084 / r268091;
        double r268093 = r268083 * r268092;
        double r268094 = 1.0;
        double r268095 = r268093 + r268094;
        double r268096 = 3.0;
        double r268097 = pow(r268095, r268096);
        double r268098 = cbrt(r268097);
        double r268099 = r268082 * r268098;
        double r268100 = sqrt(r268099);
        return r268100;
}

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
Target13.1
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 13.2

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

  3. Applied div-inv13.5

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

  5. Applied *-un-lft-identity13.5

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

  6. Applied sqrt-prod13.5

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

  7. Applied add-cube-cbrt13.5

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

  8. Applied times-frac13.5

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

  9. Simplified13.5

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

  10. Simplified13.5

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

  12. Applied add-cbrt-cube13.5

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

  13. Simplified13.5

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

  14. Final simplification13.5

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

Reproduce

herbie shell --seed 2020064 
(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))))))))