Average Error: 7.2 → 0.2
Time: 3.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re - {x.im}^{3}\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\left(3 \cdot \left(x.im \cdot x.re\right)\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r289193 = x_re;
        double r289194 = r289193 * r289193;
        double r289195 = x_im;
        double r289196 = r289195 * r289195;
        double r289197 = r289194 - r289196;
        double r289198 = r289197 * r289195;
        double r289199 = r289193 * r289195;
        double r289200 = r289195 * r289193;
        double r289201 = r289199 + r289200;
        double r289202 = r289201 * r289193;
        double r289203 = r289198 + r289202;
        return r289203;
}
↓
double f(double x_re, double x_im) {
        double r289204 = 3.0;
        double r289205 = x_im;
        double r289206 = x_re;
        double r289207 = r289205 * r289206;
        double r289208 = r289204 * r289207;
        double r289209 = r289208 * r289206;
        double r289210 = pow(r289205, r289204);
        double r289211 = r289209 - r289210;
        return r289211;
}

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

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

Derivation

  1. Initial program 7.2

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

  2. Simplified7.1

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

  4. Applied associate-*r*0.2

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

  6. Applied associate-*r*0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

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