Average Error: 2.1 → 2.1
Time: 6.3s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)
↓
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
double f(double re, double im) {
        double r16094 = 0.5;
        double r16095 = /* ERROR: no posit support in C */;
        double r16096 = 2.0;
        double r16097 = /* ERROR: no posit support in C */;
        double r16098 = re;
        double r16099 = r16098 * r16098;
        double r16100 = im;
        double r16101 = r16100 * r16100;
        double r16102 = r16099 + r16101;
        double r16103 = sqrt(r16102);
        double r16104 = r16103 + r16098;
        double r16105 = r16097 * r16104;
        double r16106 = sqrt(r16105);
        double r16107 = r16095 * r16106;
        return r16107;
}
↓
double f(double re, double im) {
        double r16108 = 0.5;
        double r16109 = 2.0;
        double r16110 = re;
        double r16111 = r16110 * r16110;
        double r16112 = im;
        double r16113 = r16112 * r16112;
        double r16114 = r16111 + r16113;
        double r16115 = sqrt(r16114);
        double r16116 = r16115 + r16110;
        double r16117 = r16109 * r16116;
        double r16118 = sqrt(r16117);
        double r16119 = r16108 * r16118;
        return r16119;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

    \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]

  2. Final simplification2.1

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (+.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))