math.cube on complex, real part

Average Error: 6.8 → 0.3

Time: 53.3s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r64831080 = x_re;
        double r64831081 = r64831080 * r64831080;
        double r64831082 = x_im;
        double r64831083 = r64831082 * r64831082;
        double r64831084 = r64831081 - r64831083;
        double r64831085 = r64831084 * r64831080;
        double r64831086 = r64831080 * r64831082;
        double r64831087 = r64831082 * r64831080;
        double r64831088 = r64831086 + r64831087;
        double r64831089 = r64831088 * r64831082;
        double r64831090 = r64831085 - r64831089;
        return r64831090;
}

↓

double f(double x_re, double x_im) {
        double r64831091 = x_re;
        double r64831092 = x_im;
        double r64831093 = r64831091 - r64831092;
        double r64831094 = r64831093 * r64831091;
        double r64831095 = r64831092 + r64831091;
        double r64831096 = r64831094 * r64831095;
        double r64831097 = r64831091 * r64831092;
        double r64831098 = r64831097 + r64831097;
        double r64831099 = r64831098 * r64831092;
        double r64831100 = r64831096 - r64831099;
        return r64831100;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.8
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 6.8

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

3. Applied difference-of-squares 6.8

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

4. Applied associate-*l* 0.3

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

5. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :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)))