Average Error: 7.3 → 0.2
Time: 3.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3}\]
\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
↓
3 \cdot \left(\left(x.im \cdot x.re\right) \cdot x.re\right) - {x.im}^{3}
double f(double x_re, double x_im) {
        double r313947 = x_re;
        double r313948 = r313947 * r313947;
        double r313949 = x_im;
        double r313950 = r313949 * r313949;
        double r313951 = r313948 - r313950;
        double r313952 = r313951 * r313949;
        double r313953 = r313947 * r313949;
        double r313954 = r313949 * r313947;
        double r313955 = r313953 + r313954;
        double r313956 = r313955 * r313947;
        double r313957 = r313952 + r313956;
        return r313957;
}
↓
double f(double x_re, double x_im) {
        double r313958 = 3.0;
        double r313959 = x_im;
        double r313960 = x_re;
        double r313961 = r313959 * r313960;
        double r313962 = r313961 * r313960;
        double r313963 = r313958 * r313962;
        double r313964 = pow(r313959, r313958);
        double r313965 = r313963 - r313964;
        return r313965;
}

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

Derivation

  1. Initial program 7.3

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

  2. Simplified7.3

    \[\leadsto \color{blue}{3 \cdot \left(x.im \cdot \left(x.re \cdot x.re\right)\right) - {x.im}^{3}}\]
  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019352 +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)))