Average Error: 0.2 → 0.2
Time: 1.6s
Precision: binary64
\[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
↓
\[x \cdot \mathsf{fma}\left(1, 6, x \cdot -9\right) + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right) \]
\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x
↓
x \cdot \mathsf{fma}\left(1, 6, x \cdot -9\right) + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right)

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))
↓
(FPCore (x)
 :precision binary64
 (+ (* x (fma 1.0 6.0 (* x -9.0))) (* x (fma (- x) 9.0 (* x 9.0)))))
double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}
↓
double code(double x) {
	return (x * fma(1.0, 6.0, (x * -9.0))) + (x * fma(-x, 9.0, (x * 9.0)));
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded in x around 0 0.2

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

  4. Applied *-un-lft-identity_binary640.2

    \[\leadsto x \cdot \left(\color{blue}{1 \cdot 6} - 9 \cdot x\right) \]

  5. Applied prod-diff_binary640.2

    \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(1, 6, -x \cdot 9\right) + \mathsf{fma}\left(-x, 9, x \cdot 9\right)\right)} \]

  6. Applied distribute-rgt-in_binary640.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, 6, -x \cdot 9\right) \cdot x + \mathsf{fma}\left(-x, 9, x \cdot 9\right) \cdot x} \]

  7. Final simplification0.2

    \[\leadsto x \cdot \mathsf{fma}\left(1, 6, x \cdot -9\right) + x \cdot \mathsf{fma}\left(-x, 9, x \cdot 9\right) \]

Reproduce

herbie shell --seed 2021313 
(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))