Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Average Error: 0.2 → 0.2

Time: 2.0s

Precision: 64

Math

\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]

↓

\[x \cdot \left(6 - 9 \cdot x\right) + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right)\]

C

double code(double x) {
	return ((3.0 * (2.0 - (x * 3.0))) * x);
}

↓

double code(double x) {
	return ((x * (6.0 - (9.0 * x))) + (x * fma(-x, 9.0, (x * 9.0))));
}

Error

Bits error versus x

Target

Original	0.2
Target	0.2
Herbie	0.2

\[6 \cdot x - 9 \cdot \left(x \cdot x\right)\]

Derivation

1. Initial program 0.2

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]

2. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{6 \cdot x - 9 \cdot {x}^{2}}\]

3. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \left(6 - 9 \cdot x\right)}\]
4. Using strategy rm

5. Applied add-cube-cbrt 1.1

   \[\leadsto x \cdot \left(\color{blue}{\left(\sqrt[3]{6} \cdot \sqrt[3]{6}\right) \cdot \sqrt[3]{6}} - 9 \cdot x\right)\]

6. Applied prod-diff 1.1

   \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{6} \cdot \sqrt[3]{6}, \sqrt[3]{6}, -x \cdot 9\right) + \mathsf{fma}\left(-x, 9, x \cdot 9\right)\right)}\]

7. Applied distribute-lft-in 1.1

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(\sqrt[3]{6} \cdot \sqrt[3]{6}, \sqrt[3]{6}, -x \cdot 9\right) + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right)}\]

8. Simplified 0.2

   \[\leadsto \color{blue}{x \cdot \left(6 - 9 \cdot x\right)} + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right)\]

9. Final simplification 0.2

   \[\leadsto x \cdot \left(6 - 9 \cdot x\right) + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right)\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6 x) (* 9 (* x x)))

  (* (* 3 (- 2 (* x 3))) x))