math.cube on complex, real part

Average Error: 6.7 → 0.7

Time: 16.6s

Precision: 64

Math

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

↓

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

C

double f(double x_re, double x_im) {
        double r2280212 = x_re;
        double r2280213 = r2280212 * r2280212;
        double r2280214 = x_im;
        double r2280215 = r2280214 * r2280214;
        double r2280216 = r2280213 - r2280215;
        double r2280217 = r2280216 * r2280212;
        double r2280218 = r2280212 * r2280214;
        double r2280219 = r2280214 * r2280212;
        double r2280220 = r2280218 + r2280219;
        double r2280221 = r2280220 * r2280214;
        double r2280222 = r2280217 - r2280221;
        return r2280222;
}

↓

double f(double x_re, double x_im) {
        double r2280223 = x_re;
        double r2280224 = x_im;
        double r2280225 = r2280223 + r2280224;
        double r2280226 = cbrt(r2280225);
        double r2280227 = r2280223 - r2280224;
        double r2280228 = r2280223 * r2280227;
        double r2280229 = r2280226 * r2280228;
        double r2280230 = r2280226 * r2280226;
        double r2280231 = r2280229 * r2280230;
        double r2280232 = r2280224 * r2280223;
        double r2280233 = r2280232 + r2280232;
        double r2280234 = r2280224 * r2280233;
        double r2280235 = r2280231 - r2280234;
        return r2280235;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original	6.7
Target	0.3
Herbie	0.7

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

Derivation

1. Initial program 6.7

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

3. Applied difference-of-squares 6.7

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

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 0.7

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

7. Applied associate-*l* 0.7

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

9. Applied *-un-lft-identity 0.7

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

10. Final simplification 0.7

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

Reproduce

herbie shell --seed 2019154 
(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)))