Average Error: 7.4 → 0.2
Time: 3.1s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[-3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.im\right) + {x.re}^{3}\]
\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
↓
-3 \cdot \left(\left(x.re \cdot x.im\right) \cdot x.im\right) + {x.re}^{3}
double f(double x_re, double x_im) {
        double r197138 = x_re;
        double r197139 = r197138 * r197138;
        double r197140 = x_im;
        double r197141 = r197140 * r197140;
        double r197142 = r197139 - r197141;
        double r197143 = r197142 * r197138;
        double r197144 = r197138 * r197140;
        double r197145 = r197140 * r197138;
        double r197146 = r197144 + r197145;
        double r197147 = r197146 * r197140;
        double r197148 = r197143 - r197147;
        return r197148;
}
↓
double f(double x_re, double x_im) {
        double r197149 = -3.0;
        double r197150 = x_re;
        double r197151 = x_im;
        double r197152 = r197150 * r197151;
        double r197153 = r197152 * r197151;
        double r197154 = r197149 * r197153;
        double r197155 = 3.0;
        double r197156 = pow(r197150, r197155);
        double r197157 = r197154 + r197156;
        return r197157;
}

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

Original7.4
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 7.4

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

  2. Simplified7.4

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

  4. Applied distribute-lft-neg-in7.4

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

  5. Applied associate-*r*0.2

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

  7. Applied fma-udef0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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)))