Linear.Quaternion:$c/ from linear-1.19.1.3, A

Average Error: 0.1 → 0.1

Time: 2.5s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(y, x, \left(z \cdot z\right) \cdot 3\right) \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))

↓

(FPCore (x y z) :precision binary64 (fma y x (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (z * z);
}

↓

double code(double x, double y, double z) {
	return fma(y, x, ((z * z) * 3.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(3 \cdot z\right) \cdot z + y \cdot x \]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

3. Taylor expanded in z around 0 0.1

   \[\leadsto \color{blue}{y \cdot x + 3 \cdot {z}^{2}} \]

4. Simplified 0.1

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

5. Applied add-cube-cbrt_binary64 0.1

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

6. Applied associate-*l*_binary64 0.2

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

7. Applied pow1_binary64 0.2

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

8. Applied pow1_binary64 0.2

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

9. Applied pow-prod-down_binary64 0.2

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

10. Applied pow1_binary64 0.2

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

11. Applied pow-prod-down_binary64 0.2

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

12. Applied pow1_binary64 0.2

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

13. Applied pow1_binary64 0.2

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

14. Applied pow-prod-down_binary64 0.2

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

15. Applied pow-prod-down_binary64 0.2

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

16. Simplified 0.1

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

17. Final simplification 0.1

    \[\leadsto \mathsf{fma}\left(y, x, \left(z \cdot z\right) \cdot 3\right) \]

Reproduce

herbie shell --seed 2022076 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))