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

Average Error: 0.3 → 0.1

Time: 2.0s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

↓

(FPCore (x) :precision binary64 (fma 6.0 x (* x (* x -9.0))))

C

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

↓

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

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. Simplified 0.2

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

3. Taylor expanded in x around 0 0.2

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

4. Applied fma-neg_binary64 0.1

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

5. Simplified 0.1

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

6. Applied associate-*r*_binary64 0.1

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022068 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))