Average Error: 6.9 → 0.2
Time: 28.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re\]
\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(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re
double f(double x_re, double x_im) {
        double r3872103 = x_re;
        double r3872104 = r3872103 * r3872103;
        double r3872105 = x_im;
        double r3872106 = r3872105 * r3872105;
        double r3872107 = r3872104 - r3872106;
        double r3872108 = r3872107 * r3872105;
        double r3872109 = r3872103 * r3872105;
        double r3872110 = r3872105 * r3872103;
        double r3872111 = r3872109 + r3872110;
        double r3872112 = r3872111 * r3872103;
        double r3872113 = r3872108 + r3872112;
        return r3872113;
}
↓
double f(double x_re, double x_im) {
        double r3872114 = x_im;
        double r3872115 = x_re;
        double r3872116 = r3872115 + r3872114;
        double r3872117 = r3872114 * r3872116;
        double r3872118 = r3872115 - r3872114;
        double r3872119 = r3872117 * r3872118;
        double r3872120 = r3872114 * r3872115;
        double r3872121 = r3872120 + r3872120;
        double r3872122 = r3872121 * r3872115;
        double r3872123 = r3872119 + r3872122;
        return r3872123;
}

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

Original6.9
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 6.9

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

  2. Taylor expanded around 0 6.8

    \[\leadsto \color{blue}{\left(x.im \cdot {x.re}^{2} - {x.im}^{3}\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019155 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :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)))