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

Average Error: 0.1 → 0.1

Time: 2.3s

Precision: 64

Math

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

↓

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

C

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

↓

double code(double x) {
	return ((((cbrt(9.0) * cbrt(9.0)) * ((x * cbrt(9.0)) * x)) + 3.0) - (12.0 * x));
}

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

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

3. Taylor expanded around 0 0.1

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

5. Applied add-cube-cbrt 0.1

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

6. Applied associate-*l* 0.1

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

8. Applied unpow2 0.1

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

9. Applied associate-*r* 0.1

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

10. Simplified 0.1

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

11. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020106 
(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)))