Average Error: 13.1 → 13.5
Time: 17.6s
Precision: 64
\[1.000000000000000006295358232172963997211 \cdot 10^{-150} \lt \left|x\right| \lt 9.999999999999999808355961724373745905731 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{\left(\sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right) \cdot \sqrt[3]{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\]
\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 \sqrt[3]{x}\right) \cdot \frac{\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)}
double f(double p, double x) {
        double r8976099 = 0.5;
        double r8976100 = 1.0;
        double r8976101 = x;
        double r8976102 = 4.0;
        double r8976103 = p;
        double r8976104 = r8976102 * r8976103;
        double r8976105 = r8976104 * r8976103;
        double r8976106 = r8976101 * r8976101;
        double r8976107 = r8976105 + r8976106;
        double r8976108 = sqrt(r8976107);
        double r8976109 = r8976101 / r8976108;
        double r8976110 = r8976100 + r8976109;
        double r8976111 = r8976099 * r8976110;
        double r8976112 = sqrt(r8976111);
        return r8976112;
}
↓
double f(double p, double x) {
        double r8976113 = 0.5;
        double r8976114 = 1.0;
        double r8976115 = x;
        double r8976116 = cbrt(r8976115);
        double r8976117 = r8976116 * r8976116;
        double r8976118 = 4.0;
        double r8976119 = p;
        double r8976120 = r8976118 * r8976119;
        double r8976121 = r8976120 * r8976119;
        double r8976122 = r8976115 * r8976115;
        double r8976123 = r8976121 + r8976122;
        double r8976124 = sqrt(r8976123);
        double r8976125 = cbrt(r8976124);
        double r8976126 = r8976125 * r8976125;
        double r8976127 = r8976126 * r8976125;
        double r8976128 = r8976116 / r8976127;
        double r8976129 = r8976117 * r8976128;
        double r8976130 = r8976114 + r8976129;
        double r8976131 = r8976113 * r8976130;
        double r8976132 = sqrt(r8976131);
        return r8976132;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.1
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 13.1

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

  3. Applied *-un-lft-identity13.1

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

  4. Applied sqrt-prod13.1

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

  5. Applied add-cube-cbrt15.0

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

  6. Applied times-frac14.9

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

  7. Simplified14.9

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

  9. Applied add-cube-cbrt13.5

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

  10. Final simplification13.5

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

Reproduce

herbie shell --seed 2019171 
(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))))))))