Average Error: 7.5 → 0.2
Time: 2.2s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im\]
\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(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.im
double f(double x_re, double x_im) {
        double r200300 = x_re;
        double r200301 = r200300 * r200300;
        double r200302 = x_im;
        double r200303 = r200302 * r200302;
        double r200304 = r200301 - r200303;
        double r200305 = r200304 * r200300;
        double r200306 = r200300 * r200302;
        double r200307 = r200302 * r200300;
        double r200308 = r200306 + r200307;
        double r200309 = r200308 * r200302;
        double r200310 = r200305 - r200309;
        return r200310;
}
↓
double f(double x_re, double x_im) {
        double r200311 = x_re;
        double r200312 = 3.0;
        double r200313 = pow(r200311, r200312);
        double r200314 = x_im;
        double r200315 = r200314 * r200311;
        double r200316 = r200312 * r200315;
        double r200317 = r200316 * r200314;
        double r200318 = r200313 - r200317;
        return r200318;
}

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

Original7.5
Target0.2
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 7.5

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  8. Applied associate-*l*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 
(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)))