Average Error: 7.3 → 0.2
Time: 24.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \left(\left(x.im \cdot 3\right) \cdot x.re\right) \cdot x.im\]
\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(x.im \cdot 3\right) \cdot x.re\right) \cdot x.im
double f(double x_re, double x_im) {
        double r231035 = x_re;
        double r231036 = r231035 * r231035;
        double r231037 = x_im;
        double r231038 = r231037 * r231037;
        double r231039 = r231036 - r231038;
        double r231040 = r231039 * r231035;
        double r231041 = r231035 * r231037;
        double r231042 = r231037 * r231035;
        double r231043 = r231041 + r231042;
        double r231044 = r231043 * r231037;
        double r231045 = r231040 - r231044;
        return r231045;
}
↓
double f(double x_re, double x_im) {
        double r231046 = x_re;
        double r231047 = 3.0;
        double r231048 = pow(r231046, r231047);
        double r231049 = x_im;
        double r231050 = r231049 * r231047;
        double r231051 = r231050 * r231046;
        double r231052 = r231051 * r231049;
        double r231053 = r231048 - r231052;
        return r231053;
}

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

Original7.3
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 7.3

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

  2. Simplified0.2

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 
(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)))