math.cube on complex, imaginary part

Average Error: 6.7 → 0.2

Time: 21.4s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r8741081 = x_re;
        double r8741082 = r8741081 * r8741081;
        double r8741083 = x_im;
        double r8741084 = r8741083 * r8741083;
        double r8741085 = r8741082 - r8741084;
        double r8741086 = r8741085 * r8741083;
        double r8741087 = r8741081 * r8741083;
        double r8741088 = r8741083 * r8741081;
        double r8741089 = r8741087 + r8741088;
        double r8741090 = r8741089 * r8741081;
        double r8741091 = r8741086 + r8741090;
        return r8741091;
}

↓

double f(double x_re, double x_im) {
        double r8741092 = x_re;
        double r8741093 = 3.0;
        double r8741094 = x_im;
        double r8741095 = r8741093 * r8741094;
        double r8741096 = r8741095 * r8741092;
        double r8741097 = r8741092 * r8741096;
        double r8741098 = pow(r8741094, r8741093);
        double r8741099 = r8741097 - r8741098;
        return r8741099;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.7
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 6.7

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

2. Taylor expanded around 0 6.7

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

3. Simplified 0.2

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

5. Applied pow1 0.2

   \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(x.im \cdot x.im\right) \cdot \color{blue}{{x.im}^{1}}\]

6. Applied pow1 0.2

   \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(x.im \cdot \color{blue}{{x.im}^{1}}\right) \cdot {x.im}^{1}\]

7. Applied pow1 0.2

   \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - \left(\color{blue}{{x.im}^{1}} \cdot {x.im}^{1}\right) \cdot {x.im}^{1}\]

8. Applied pow-prod-up 0.2

   \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{\left(1 + 1\right)}} \cdot {x.im}^{1}\]

9. Applied pow-prod-up 0.2

   \[\leadsto \left(\left(3 \cdot x.im\right) \cdot x.re\right) \cdot x.re - \color{blue}{{x.im}^{\left(\left(1 + 1\right) + 1\right)}}\]

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019168 
(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)))