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

Average Error: 0.1 → 0.1

Time: 1.9s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))

↓

(FPCore (x) :precision binary64 (- (fma 9.0 (* x x) 3.0) (* x 12.0)))

C

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

↓

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

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

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

3. Taylor expanded in x around 0 0.1

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

4. Applied add-cube-cbrt_binary64 0.1

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

5. Applied associate-*l*_binary64 0.1

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

6. Taylor expanded in x around 0 0.1

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

7. Simplified 0.1

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

8. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))