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

Average Error: 0.2 → 0.1

Time: 13.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r465901 = 3.0;
        double r465902 = x;
        double r465903 = r465902 * r465901;
        double r465904 = r465903 * r465902;
        double r465905 = 4.0;
        double r465906 = r465902 * r465905;
        double r465907 = r465904 - r465906;
        double r465908 = 1.0;
        double r465909 = r465907 + r465908;
        double r465910 = r465901 * r465909;
        return r465910;
}

↓

double f(double x) {
        double r465911 = 3.0;
        double r465912 = x;
        double r465913 = 9.0;
        double r465914 = r465913 * r465912;
        double r465915 = 12.0;
        double r465916 = r465914 - r465915;
        double r465917 = r465912 * r465916;
        double r465918 = r465911 + r465917;
        return r465918;
}

Error

Bits error versus x

Target

Original	0.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

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

3. Taylor expanded around 0 0.1

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

4. Simplified 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019209 +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)))