Average Error: 12.6 → 13.2
Time: 4.8s
Precision: binary64
\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}} \cdot \sqrt[3]{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \left(\sqrt[3]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}} \cdot \sqrt[3]{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}\right) \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}\right)}
(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))
↓
(FPCore (p x)
 :precision binary64
 (sqrt
  (*
   0.5
   (+
    1.0
    (*
     (*
      (cbrt x)
      (/
       (cbrt x)
       (*
        (cbrt (sqrt (+ (* 4.0 (* p p)) (* x x))))
        (cbrt (sqrt (+ (* 4.0 (* p p)) (* x x)))))))
     (/ (cbrt x) (cbrt (sqrt (+ (* 4.0 (* p p)) (* x x))))))))))
double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))));
}
↓
double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (((double) (((double) cbrt(x)) * (((double) cbrt(x)) / ((double) (((double) cbrt(((double) sqrt(((double) (((double) (4.0 * ((double) (p * p)))) + ((double) (x * x)))))))) * ((double) cbrt(((double) sqrt(((double) (((double) (4.0 * ((double) (p * p)))) + ((double) (x * x))))))))))))) * (((double) cbrt(x)) / ((double) cbrt(((double) sqrt(((double) (((double) (4.0 * ((double) (p * p)))) + ((double) (x * x)))))))))))))))));
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.6
Target12.6
Herbie13.2
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 12.6

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

  3. Applied add-cube-cbrt14.6

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

  4. Applied add-cube-cbrt12.7

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

  5. Applied times-frac12.7

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

  6. Simplified13.2

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

  7. Simplified13.2

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

  8. Final simplification13.2

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

Reproduce

herbie shell --seed 2020198 
(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 1.0 (/ (* 2.0 p) x)))))

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