simple fma test

Average Error: 44.7 → 8.1

Time: 24.4s

Precision: 64

Math

\[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]

↓

\[\left(\left(\left(\left(\log \left(\frac{e^{\mathsf{fma}\left(x, y, z\right) - \left(z + x \cdot y\right)}}{e}\right)\right)\right)\right)\right)\]

C

double f(double x, double y, double z) {
        double r2966250 = x;
        double r2966251 = y;
        double r2966252 = z;
        double r2966253 = fma(r2966250, r2966251, r2966252);
        double r2966254 = 1.0;
        double r2966255 = r2966250 * r2966251;
        double r2966256 = r2966255 + r2966252;
        double r2966257 = r2966254 + r2966256;
        double r2966258 = r2966253 - r2966257;
        return r2966258;
}

↓

double f(double x, double y, double z) {
        double r2966259 = x;
        double r2966260 = y;
        double r2966261 = z;
        double r2966262 = fma(r2966259, r2966260, r2966261);
        double r2966263 = r2966259 * r2966260;
        double r2966264 = r2966261 + r2966263;
        double r2966265 = r2966262 - r2966264;
        double r2966266 = exp(r2966265);
        double r2966267 = exp(1.0);
        double r2966268 = r2966266 / r2966267;
        double r2966269 = log(r2966268);
        double r2966270 = /* ERROR: no posit support in C */;
        double r2966271 = /* ERROR: no posit support in C */;
        double r2966272 = /* ERROR: no posit support in C */;
        double r2966273 = /* ERROR: no posit support in C */;
        return r2966273;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	44.7
Target	0
Herbie	8.1

\[-1\]

Derivation

1. Initial program 44.7

   \[\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\]
2. Using strategy rm

3. Applied insert-posit16 44.7

   \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(x \cdot y + z\right)\right)\right)\right)}\]
4. Using strategy rm

5. Applied add-cube-cbrt 44.0

   \[\leadsto \left(\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \color{blue}{\left(\sqrt[3]{x \cdot y + z} \cdot \sqrt[3]{x \cdot y + z}\right) \cdot \sqrt[3]{x \cdot y + z}}\right)\right)\right)\]
6. Using strategy rm

7. Applied insert-posit16 44.0

   \[\leadsto \left(\color{blue}{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - \left(1 + \left(\sqrt[3]{x \cdot y + z} \cdot \sqrt[3]{x \cdot y + z}\right) \cdot \sqrt[3]{x \cdot y + z}\right)\right)\right)\right)}\right)\]

8. Simplified 33.5

   \[\leadsto \left(\left(\color{blue}{\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - x \cdot y\right) - \left(1 + z\right)\right)\right)}\right)\right)\]
9. Using strategy rm

10. Applied add-log-exp 35.1

    \[\leadsto \left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - x \cdot y\right) - \left(1 + \color{blue}{\log \left(e^{z}\right)}\right)\right)\right)\right)\right)\]

11. Applied add-log-exp 35.1

    \[\leadsto \left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - x \cdot y\right) - \left(\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{z}\right)\right)\right)\right)\right)\right)\]

12. Applied sum-log 35.1

    \[\leadsto \left(\left(\left(\left(\left(\mathsf{fma}\left(x, y, z\right) - x \cdot y\right) - \color{blue}{\log \left(e^{1} \cdot e^{z}\right)}\right)\right)\right)\right)\]

13. Applied add-log-exp 35.1

    \[\leadsto \left(\left(\left(\left(\color{blue}{\log \left(e^{\mathsf{fma}\left(x, y, z\right) - x \cdot y}\right)} - \log \left(e^{1} \cdot e^{z}\right)\right)\right)\right)\right)\]

14. Applied diff-log 35.1

    \[\leadsto \left(\left(\left(\color{blue}{\left(\log \left(\frac{e^{\mathsf{fma}\left(x, y, z\right) - x \cdot y}}{e^{1} \cdot e^{z}}\right)\right)}\right)\right)\right)\]

15. Simplified 8.1

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

16. Final simplification 8.1

    \[\leadsto \left(\left(\left(\left(\log \left(\frac{e^{\mathsf{fma}\left(x, y, z\right) - \left(z + x \cdot y\right)}}{e}\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x y z)
  :name "simple fma test"

  :herbie-target
  -1

  (- (fma x y z) (+ 1 (+ (* x y) z))))