Average Error: 7.9 → 0.2
Time: 6.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
{x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r203161 = x_re;
        double r203162 = r203161 * r203161;
        double r203163 = x_im;
        double r203164 = r203163 * r203163;
        double r203165 = r203162 - r203164;
        double r203166 = r203165 * r203161;
        double r203167 = r203161 * r203163;
        double r203168 = r203163 * r203161;
        double r203169 = r203167 + r203168;
        double r203170 = r203169 * r203163;
        double r203171 = r203166 - r203170;
        return r203171;
}
↓
double f(double x_re, double x_im) {
        double r203172 = x_re;
        double r203173 = 3.0;
        double r203174 = pow(r203172, r203173);
        double r203175 = r203173 * r203172;
        double r203176 = x_im;
        double r203177 = r203175 * r203176;
        double r203178 = r203177 * r203176;
        double r203179 = r203174 - r203178;
        return r203179;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
Target0.3
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.9

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]

  2. Simplified0.2

    \[\leadsto \color{blue}{{x.re}^{3} - 3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.im\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.2

    \[\leadsto {x.re}^{3} - \color{blue}{\left(3 \cdot \left(x.re \cdot x.im\right)\right) \cdot x.im}\]
  5. Using strategy rm

  6. Applied associate-*r*0.2

    \[\leadsto {x.re}^{3} - \color{blue}{\left(\left(3 \cdot x.re\right) \cdot x.im\right)} \cdot x.im\]

  7. Final simplification0.2

    \[\leadsto {x.re}^{3} - \left(\left(3 \cdot x.re\right) \cdot x.im\right) \cdot x.im\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

  (- (* (- (* x.re x.re) (* x.im x.im)) x.re) (* (+ (* x.re x.im) (* x.im x.re)) x.im)))