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

Average Error: 0.3 → 0.1

Time: 2.7s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(6, x, -9 \cdot {x}^{2}\right)\]

C

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

↓

double code(double x) {
	return fma(6.0, x, -(9.0 * pow(x, 2.0)));
}

Error

Bits error versus x

Target

Original	0.3
Target	0.2
Herbie	0.1

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

Derivation

1. Initial program 0.3

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x\]
2. Using strategy rm

3. Applied flip-- 0.3

   \[\leadsto \left(3 \cdot \color{blue}{\frac{2 \cdot 2 - \left(x \cdot 3\right) \cdot \left(x \cdot 3\right)}{2 + x \cdot 3}}\right) \cdot x\]

4. Simplified 0.3

   \[\leadsto \left(3 \cdot \frac{\color{blue}{\mathsf{fma}\left(3, x, 2\right) \cdot \left(2 - 3 \cdot x\right)}}{2 + x \cdot 3}\right) \cdot x\]

5. Simplified 0.3

   \[\leadsto \left(3 \cdot \frac{\mathsf{fma}\left(3, x, 2\right) \cdot \left(2 - 3 \cdot x\right)}{\color{blue}{\mathsf{fma}\left(3, x, 2\right)}}\right) \cdot x\]

6. Taylor expanded around 0 0.2

   \[\leadsto \color{blue}{6 \cdot x - 9 \cdot {x}^{2}}\]
7. Using strategy rm

8. Applied fma-neg 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(6, x, -9 \cdot {x}^{2}\right)}\]

9. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(6, x, -9 \cdot {x}^{2}\right)\]

Reproduce

herbie shell --seed 2020057 +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))