math.cube on complex, real part

Average Error: 7.0 → 0.5

Time: 27.1s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r3507102 = x_re;
        double r3507103 = r3507102 * r3507102;
        double r3507104 = x_im;
        double r3507105 = r3507104 * r3507104;
        double r3507106 = r3507103 - r3507105;
        double r3507107 = r3507106 * r3507102;
        double r3507108 = r3507102 * r3507104;
        double r3507109 = r3507104 * r3507102;
        double r3507110 = r3507108 + r3507109;
        double r3507111 = r3507110 * r3507104;
        double r3507112 = r3507107 - r3507111;
        return r3507112;
}

↓

double f(double x_re, double x_im) {
        double r3507113 = x_re;
        double r3507114 = x_im;
        double r3507115 = r3507113 - r3507114;
        double r3507116 = r3507115 * r3507113;
        double r3507117 = r3507114 + r3507113;
        double r3507118 = r3507116 * r3507117;
        double r3507119 = cbrt(r3507114);
        double r3507120 = r3507113 * r3507114;
        double r3507121 = r3507120 + r3507120;
        double r3507122 = r3507121 * r3507119;
        double r3507123 = r3507122 * r3507119;
        double r3507124 = r3507119 * r3507123;
        double r3507125 = r3507118 - r3507124;
        return r3507125;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	7.0
Target	0.2
Herbie	0.5

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

Derivation

1. Initial program 7.0

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

3. Applied difference-of-squares 7.0

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

4. Applied associate-*l* 0.2

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

6. Applied add-cube-cbrt 0.5

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

7. Applied associate-*r* 0.5

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

8. Simplified 0.5

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

9. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019141 
(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)))