SynthBasics:oscSampleBasedAux from YampaSynth-0.2

Average Error: 0.0 → 0.0

Time: 9.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x + y \cdot \left(z - x\right)\]

↓

\[\mathsf{fma}\left(z, y, x\right) - x \cdot y\]

C

double f(double x, double y, double z) {
        double r12683 = x;
        double r12684 = y;
        double r12685 = z;
        double r12686 = r12685 - r12683;
        double r12687 = r12684 * r12686;
        double r12688 = r12683 + r12687;
        return r12688;
}

↓

double f(double x, double y, double z) {
        double r12689 = z;
        double r12690 = y;
        double r12691 = x;
        double r12692 = fma(r12689, r12690, r12691);
        double r12693 = r12691 * r12690;
        double r12694 = r12692 - r12693;
        return r12694;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Initial program 0.0

   \[x + y \cdot \left(z - x\right)\]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - x, y, x\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 31.8

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z - x, y, x\right)} \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}}\]
5. Using strategy rm

6. Applied add-sqr-sqrt 31.8

   \[\leadsto \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(z - x, y, x\right)} \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}}} \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]

7. Applied sqrt-prod 31.9

   \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}\right)} \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]
8. Using strategy rm

9. Applied add-sqr-sqrt 31.9

   \[\leadsto \left(\sqrt{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(z - x, y, x\right)} \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}\right) \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]

10. Applied sqrt-prod 31.9

    \[\leadsto \left(\sqrt{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}\right) \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]

11. Applied sqrt-prod 32.0

    \[\leadsto \left(\color{blue}{\left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}} \cdot \sqrt{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}}\right)} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}\right) \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]

12. Applied associate-*l* 32.0

    \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}} \cdot \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]

13. Simplified 32.1

    \[\leadsto \left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}} \cdot \color{blue}{{\left(\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(z - x, y, x\right)}}}\right)}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(z - x, y, x\right)}\]

14. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(z \cdot y + x\right) - x \cdot y}\]

15. Simplified 0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, x\right) - x \cdot y}\]

16. Final simplification 0.0

    \[\leadsto \mathsf{fma}\left(z, y, x\right) - x \cdot y\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z)
  :name "SynthBasics:oscSampleBasedAux from YampaSynth-0.2"
  :precision binary64
  (+ x (* y (- z x))))