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

Average Error: 0.1 → 0.1

Time: 1.9s

Precision: 64

Math

\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

↓

\[\mathsf{fma}\left(x, \mathsf{fma}\left(9, x, -12\right), 3\right)\]

C

double code(double x) {
	return (3.0 * ((((x * 3.0) * x) - (x * 4.0)) + 1.0));
}

↓

double code(double x) {
	return fma(x, fma(9.0, x, -12.0), 3.0);
}

Error

Bits error versus x

Target

Original	0.1
Target	0.1
Herbie	0.1

\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

1. Initial program 0.1

   \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 3 - 4, 1\right) \cdot 3}\]

3. Taylor expanded around 0 0.1

   \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

4. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)}\]

5. Taylor expanded around 0 0.1

   \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

6. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 9 \cdot x - 12, 3\right)}\]
7. Using strategy rm

8. Applied fma-neg 0.1

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(9, x, -12\right)}, 3\right)\]

9. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(9, x, -12\right), 3\right)\]

Reproduce

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3 (- (* (* 9 x) x) (* 12 x)))

  (* 3 (+ (- (* (* x 3) x) (* x 4)) 1)))