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

Average Error: 0.1 → 0.1

Time: 19.4s

Precision: 64

Math

\[3.0 \cdot \left(\left(\left(x \cdot 3.0\right) \cdot x - x \cdot 4.0\right) + 1.0\right)\]

↓

\[\mathsf{fma}\left(9.0, x \cdot x, 3.0\right) - 12.0 \cdot x\]

C

double f(double x) {
        double r33407369 = 3.0;
        double r33407370 = x;
        double r33407371 = r33407370 * r33407369;
        double r33407372 = r33407371 * r33407370;
        double r33407373 = 4.0;
        double r33407374 = r33407370 * r33407373;
        double r33407375 = r33407372 - r33407374;
        double r33407376 = 1.0;
        double r33407377 = r33407375 + r33407376;
        double r33407378 = r33407369 * r33407377;
        return r33407378;
}

↓

double f(double x) {
        double r33407379 = 9.0;
        double r33407380 = x;
        double r33407381 = r33407380 * r33407380;
        double r33407382 = 3.0;
        double r33407383 = fma(r33407379, r33407381, r33407382);
        double r33407384 = 12.0;
        double r33407385 = r33407384 * r33407380;
        double r33407386 = r33407383 - r33407385;
        return r33407386;
}

Error

Bits error versus x

Target

Original	0.1
Target	0.1
Herbie	0.1

\[3.0 + \left(\left(9.0 \cdot x\right) \cdot x - 12.0 \cdot x\right)\]

Derivation

1. Initial program 0.1

   \[3.0 \cdot \left(\left(\left(x \cdot 3.0\right) \cdot x - x \cdot 4.0\right) + 1.0\right)\]

2. Taylor expanded around 0 0.1

   \[\leadsto \color{blue}{\left(9.0 \cdot {x}^{2} + 3.0\right) - 12.0 \cdot x}\]

3. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(9.0, x \cdot x, 3.0\right) - x \cdot 12.0}\]

4. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(9.0, x \cdot x, 3.0\right) - 12.0 \cdot x\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"

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

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