Average Error: 0.3 → 0.1
Time: 1.9s
Precision: binary64
\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
↓
\[\mathsf{fma}\left(6, x, -9 \cdot \left(x \cdot x\right)\right) \]
\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x
↓
\mathsf{fma}\left(6, x, -9 \cdot \left(x \cdot x\right)\right)

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))
↓
(FPCore (x) :precision binary64 (fma 6.0 x (* -9.0 (* x x))))
double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}
↓
double code(double x) {
	return fma(6.0, x, (-9.0 * (x * x)));
}

Error

Bits error versus x

Target

Original0.3
Target0.2
Herbie0.1
\[6 \cdot x - 9 \cdot \left(x \cdot x\right) \]

Derivation

  1. Initial program 0.3

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

  2. Simplified0.2

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

  3. Taylor expanded in x around 0 0.2

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

  4. Applied fma-neg_binary640.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2022067 
(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))