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

↓

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

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

↓

\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)

double f(double x) {
        double r786396 = 3.0;
        double r786397 = x;
        double r786398 = r786397 * r786396;
        double r786399 = r786398 * r786397;
        double r786400 = 4.0;
        double r786401 = r786397 * r786400;
        double r786402 = r786399 - r786401;
        double r786403 = 1.0;
        double r786404 = r786402 + r786403;
        double r786405 = r786396 * r786404;
        return r786405;
}

↓

double f(double x) {
        double r786406 = x;
        double r786407 = 2.0;
        double r786408 = pow(r786406, r786407);
        double r786409 = 9.0;
        double r786410 = 3.0;
        double r786411 = 12.0;
        double r786412 = r786411 * r786406;
        double r786413 = r786410 - r786412;
        double r786414 = fma(r786408, r786409, r786413);
        return r786414;
}

Original	0.2
Target	0.1
Herbie	0.1

Derivation

Initial program 0.2
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
Simplified0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 3 - 4, 1\right) \cdot 3}\]
Taylor expanded around 0 0.1
\[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]
Simplified0.1
\[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)}\]
Final simplification0.1
\[\leadsto \mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)\]

Reproduce

herbie shell --seed 2020100 +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)))

Error

Target

Derivation

Reproduce