Average Error: 7.4 → 0.3
Time: 3.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(x.im \cdot x.re\right)\right)\right) \cdot x.re - {x.im}^{3}\]
\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(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(x.im \cdot x.re\right)\right)\right) \cdot x.re - {x.im}^{3}
double f(double x_re, double x_im) {
        double r242224 = x_re;
        double r242225 = r242224 * r242224;
        double r242226 = x_im;
        double r242227 = r242226 * r242226;
        double r242228 = r242225 - r242227;
        double r242229 = r242228 * r242226;
        double r242230 = r242224 * r242226;
        double r242231 = r242226 * r242224;
        double r242232 = r242230 + r242231;
        double r242233 = r242232 * r242224;
        double r242234 = r242229 + r242233;
        return r242234;
}
↓
double f(double x_re, double x_im) {
        double r242235 = 3.0;
        double r242236 = cbrt(r242235);
        double r242237 = r242236 * r242236;
        double r242238 = x_im;
        double r242239 = x_re;
        double r242240 = r242238 * r242239;
        double r242241 = r242236 * r242240;
        double r242242 = r242237 * r242241;
        double r242243 = r242242 * r242239;
        double r242244 = pow(r242238, r242235);
        double r242245 = r242243 - r242244;
        return r242245;
}

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

Derivation

  1. Initial program 7.4

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

  2. Simplified7.4

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied associate-*l*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 +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)))